{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# AMC Preprocessing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Load Numina and AIME Problems"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Extract AMC Data from numina\n",
    "# Load AIME dataset in raw/train/aime.json and raw/test/aime.json\n",
    "import json\n",
    "from datasets import load_dataset\n",
    "\n",
    "ds = load_dataset(\"AI-MO/NuminaMath-CoT\")\n",
    "# Filter for amc_aime problems\n",
    "amc_aime = ds['train'].filter(lambda x: x['source'] == 'amc_aime')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Training dataset loaded, size: 975\n",
      "Test dataset loaded, size: 30\n",
      "1005\n"
     ]
    }
   ],
   "source": [
    "from deepscaler.data import load_dataset, TrainDataset, TestDataset\n",
    "\n",
    "train_dataset = load_dataset(TrainDataset.AIME)\n",
    "test_dataset = load_dataset(TestDataset.AIME)\n",
    "print(\"Training dataset loaded, size:\", len(train_dataset))\n",
    "print(\"Test dataset loaded, size:\", len(test_dataset))\n",
    "aime_dataset = train_dataset + test_dataset"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Filter AMC-only Problems from Numina AMC_AIME Category"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "from deepscaler.utils import RAG\n",
    "rag_server = RAG(docs=[d['problem'] for d in aime_dataset])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.9696745872497559\n",
      "Patio blocks that are hexagons $1$ unit on a side are used to outline a garden by placing the blocks edge to edge with $n$ on each side. The diagram indicates the path of blocks around the garden when $n=5$.\n",
      "[AIME 2002 II Problem 4.gif](https://artofproblemsolving.com/wiki/index.php/File:AIME_2002_II_Problem_4.gif)\n",
      "If $n=202$, then the area of the garden enclosed by the path, not including the path itself, is $m\\left(\\sqrt3/2\\right)$ square units, where $m$ is a positive integer. Find the remainder when $m$ is divided by $1000$.\n",
      "Found similar problem: Patio blocks that are hexagons $1$ unit on a side are used to outline a garden by placing the blocks edge to edge with $n$ on each side. The diagram indicates the path of blocks around the garden when $n=5$.\n",
      " If $n=202$, then the area of the garden enclosed by the path, not including the path itself, is $m\\left(\\sqrt3/2\\right)$ square units, where $m$ is a positive integer. Find the remainder when $m$ is divided by $1000$.\n",
      "\n",
      "1.0000001192092896\n",
      "Misha rolls a standard, fair six-sided die until she rolls $1-2-3$ in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is $\\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n",
      "Found similar problem: Misha rolls a standard, fair six-sided die until she rolls $1-2-3$ in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is $\\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n",
      "\n",
      "1.0000001192092896\n",
      "Diameter $AB$ of a circle has length a $2$-digit integer (base ten). Reversing the digits gives the length of the perpendicular chord $CD$. The distance from their intersection point $H$ to the center $O$ is a positive rational number. Determine the length of $AB$.\n",
      "Found similar problem: Diameter $AB$ of a circle has length a $2$-digit integer (base ten). Reversing the digits gives the length of the perpendicular chord $CD$. The distance from their intersection point $H$ to the center $O$ is a positive rational number. Determine the length of $AB$.\n",
      "\n",
      "1.0\n",
      "Let $m$ be the least positive integer divisible by $17$ whose digits sum to $17$. Find $m$.\n",
      "Found similar problem: Let $m$ be the least positive integer divisible by $17$ whose digits sum to $17$. Find $m$.\n",
      "\n",
      "1.0000001192092896\n",
      "Five towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing through other towns on the way).\n",
      "Found similar problem: Five towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing through other towns on the way).\n",
      "\n",
      "0.9686546325683594\n",
      "In the diagram below, angle $ABC$ is a right angle. Point $D$ is on $\\overline{BC}$, and $\\overline{AD}$ bisects angle $CAB$. Points $E$ and $F$ are on $\\overline{AB}$ and $\\overline{AC}$, respectively, so that $AE=3$ and $AF=10$. Given that $EB=9$ and $FC=27$, find the integer closest to the area of quadrilateral $DCFG$.\n",
      "\n",
      "[AIME 2002I Problem 10.png](https://artofproblemsolving.com/wiki/index.php/File:AIME_2002I_Problem_10.png)\n",
      "Found similar problem: In the diagram below, angle $ABC$ is a right angle. Point $D$ is on $\\overline{BC}$, and $\\overline{AD}$ bisects angle $CAB$. Points $E$ and $F$ are on $\\overline{AB}$ and $\\overline{AC}$, respectively, so that $AE=3$ and $AF=10$. Given that $EB=9$ and $FC=27$, find the integer closest to the area of quadrilateral $DCFG$.\n",
      "\n",
      "1.0\n",
      "Squares $ABCD$ and $EFGH$ have a common center and $\\overline{AB} || \\overline{EF}$. The area of $ABCD$ is 2016, and the area of $EFGH$ is a smaller positive integer. Square $IJKL$ is constructed so that each of its vertices lies on a side of $ABCD$ and each vertex of $EFGH$ lies on a side of $IJKL$. Find the difference between the largest and smallest positive integer values for the area of $IJKL$.\n",
      "Found similar problem: Squares $ABCD$ and $EFGH$ have a common center and $\\overline{AB} || \\overline{EF}$. The area of $ABCD$ is 2016, and the area of $EFGH$ is a smaller positive integer. Square $IJKL$ is constructed so that each of its vertices lies on a side of $ABCD$ and each vertex of $EFGH$ lies on a side of $IJKL$. Find the difference between the largest and smallest positive integer values for the area of $IJKL$.\n",
      "\n",
      "0.999163031578064\n",
      "For each positive integer $n$, let $f(n) = \\sum_{k = 1}^{100} \\lfloor \\log_{10} (kn) \\rfloor$. Find the largest value of $n$ for which $f(n) \\le 300$.\n",
      "Note: $\\lfloor x \\rfloor$ is the greatest integer less than or equal to $x$.\n",
      "Found similar problem: For each positive integer n, let $f(n) = \\sum_{k = 1}^{100} \\lfloor \\log_{10} (kn) \\rfloor$. Find the largest value of $n$ for which $f(n) \\le 300$.\n",
      "Note: $\\lfloor x \\rfloor$ is the greatest integer less than or equal to $x$.\n",
      "\n",
      "0.9873483180999756\n",
      "Let $P$ be an interior point of triangle $ABC$ and extend lines from the vertices through $P$ to the opposite sides.  Let $a$, $b$, $c$, and $d$ denote the lengths of the segments indicated in the figure.  Find the product $abc$ if $a + b + c = 43$ and $d = 3$.\n",
      "[1988 AIME-12.png](https://artofproblemsolving.com/wiki/index.php/File:1988_AIME-12.png)\n",
      "Found similar problem: Let $P$ be an interior point of triangle $ABC$ and extend lines from the vertices through $P$ to the opposite sides. Let $a$, $b$, $c$, and $d$ denote the lengths of the segments indicated in the figure. Find the product $abc$ if $a + b + c = 43$ and $d = 3$.\n",
      "\n",
      "0.8249955177307129\n",
      "In a game of Chomp, two players alternately take bites from a 5-by-7 grid of [unit squares](https://artofproblemsolving.com/wiki/index.php/Unit_square). To take a bite, a player chooses one of the remaining [ squares](https://artofproblemsolving.com/wiki/index.php/Square_(geometry)), then removes (\"eats\") all squares in the quadrant defined by the left edge (extended upward) and the lower edge (extended rightward) of the chosen square. For example, the bite determined by the shaded square in the diagram would remove the shaded square and the four squares marked by $\\times.$ (The squares with two or more dotted edges have been removed form the original board in previous moves.) \n",
      "[AIME 1992 Problem 12.png](https://artofproblemsolving.com/wiki/index.php/File:AIME_1992_Problem_12.png)\n",
      "The object of the game is to make one's opponent take the last bite. The diagram shows one of the many [subsets](https://artofproblemsolving.com/wiki/index.php/Subset) of the [set](https://artofproblemsolving.com/wiki/index.php/Set) of 35 unit squares that can occur during the game of Chomp. How many different subsets are there in all? Include the full board and empty board in your count.\n",
      "Found similar problem: In a game of Chomp, two players alternately take bites from a 5-by-7 grid of unit squares. To take a bite, a player chooses one of the remaining squares, then removes (\"eats\") all squares in the quadrant defined by the left edge (extended upward) and the lower edge (extended rightward) of the chosen square. For example, the bite determined by the shaded square in the diagram would remove the shaded square and the four squares marked by $\\times.$ (The squares with two or more dotted edges have been removed form the original board in previous moves.)\n",
      " The object of the game is to make one's opponent take the last bite. The diagram shows one of the many subsets of the set of 35 unit squares that can occur during the game of Chomp. How many different subsets are there in all? Include the full board and empty board in your count.\n",
      "\n",
      "1.0\n",
      "The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.\n",
      "[asy] draw(Circle((0,0), 4)); draw(Circle((0,0), 3)); draw((0,4)--(0,3)); draw((0,-4)--(0,-3)); draw((-2.598, 1.5)--(-3.4641, 2)); draw((-2.598, -1.5)--(-3.4641, -2)); draw((2.598, -1.5)--(3.4641, -2)); draw((2.598, 1.5)--(3.4641, 2)); [/asy]\n",
      "Found similar problem: The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color.\n",
      "[asy] draw(Circle((0,0), 4)); draw(Circle((0,0), 3)); draw((0,4)--(0,3)); draw((0,-4)--(0,-3)); draw((-2.598, 1.5)--(-3.4641, 2)); draw((-2.598, -1.5)--(-3.4641, -2)); draw((2.598, -1.5)--(3.4641, -2)); draw((2.598, 1.5)--(3.4641, 2)); [/asy]\n",
      "\n",
      "1.0\n",
      "Find the sum of all positive integers $a=2^n3^m$ where $n$ and $m$ are non-negative integers, for which $a^6$ is not a divisor of $6^a$.\n",
      "Found similar problem: Find the sum of all positive integers $a=2^n3^m$ where $n$ and $m$ are non-negative integers, for which $a^6$ is not a divisor of $6^a$.\n",
      "\n",
      "1.0000001192092896\n",
      "Let $A_1A_2A_3\\ldots A_{12}$ be a dodecagon ($12$-gon). Three frogs initially sit at $A_4,A_8,$ and $A_{12}$. At the end of each minute, simultaneously, each of the three frogs jumps to one of the two vertices adjacent to its current position, chosen randomly and independently with both choices being equally likely. All three frogs stop jumping as soon as two frogs arrive at the same vertex at the same time. The expected number of minutes until the frogs stop jumping is $\\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n",
      "Found similar problem: Let $A_1A_2A_3\\ldots A_{12}$ be a dodecagon ($12$-gon). Three frogs initially sit at $A_4,A_8,$ and $A_{12}$. At the end of each minute, simultaneously, each of the three frogs jumps to one of the two vertices adjacent to its current position, chosen randomly and independently with both choices being equally likely. All three frogs stop jumping as soon as two frogs arrive at the same vertex at the same time. The expected number of minutes until the frogs stop jumping is $\\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n",
      "\n",
      "1.0\n",
      "Polyhedron $ABCDEFG$ has six faces.  Face $ABCD$ is a square with $AB = 12;$ face $ABFG$ is a trapezoid with $\\overline{AB}$ parallel to $\\overline{GF},$ $BF = AG = 8,$ and $GF = 6;$ and face $CDE$ has $CE = DE = 14.$  The other three faces are $ADEG, BCEF,$ and $EFG.$  The distance from $E$ to face $ABCD$ is 12.  Given that $EG^2 = p - q\\sqrt {r},$ where $p, q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p + q + r.$\n",
      "Found similar problem: Polyhedron $ABCDEFG$ has six faces. Face $ABCD$ is a square with $AB = 12;$ face $ABFG$ is a trapezoid with $\\overline{AB}$ parallel to $\\overline{GF},$ $BF = AG = 8,$ and $GF = 6;$ and face $CDE$ has $CE = DE = 14.$ The other three faces are $ADEG, BCEF,$ and $EFG.$ The distance from $E$ to face $ABCD$ is 12. Given that $EG^2 = p - q\\sqrt {r},$ where $p, q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p + q + r.$\n",
      "\n",
      "1.0\n",
      "During a recent campaign for office, a candidate made a tour of a country which we assume lies in a plane. On the first day of the tour he went east, on the second day he went north, on the third day west, on the fourth day south, on the fifth day east, etc. If the candidate went $n^{2}_{}/2$ miles on the $n^{\\mbox{th}}_{}$ day of this tour, how many miles was he from his starting point at the end of the $40^{\\mbox{th}}_{}$ day?\n",
      "Found similar problem: During a recent campaign for office, a candidate made a tour of a country which we assume lies in a plane. On the first day of the tour he went east, on the second day he went north, on the third day west, on the fourth day south, on the fifth day east, etc. If the candidate went $n^{2}_{}/2$ miles on the $n^{\\mbox{th}}_{}$ day of this tour, how many miles was he from his starting point at the end of the $40^{\\mbox{th}}_{}$ day?\n",
      "\n",
      "1.0\n",
      "Let $m$ be the number of solutions in positive integers to the equation $4x+3y+2z=2009$, and let $n$ be the number of solutions in positive integers to the equation $4x+3y+2z=2000$. Find the remainder when $m-n$ is divided by $1000$.\n",
      "Found similar problem: Let $m$ be the number of solutions in positive integers to the equation $4x+3y+2z=2009$, and let $n$ be the number of solutions in positive integers to the equation $4x+3y+2z=2000$. Find the remainder when $m-n$ is divided by $1000$.\n",
      "\n",
      "1.0\n",
      "Find the number of functions $f$ from $\\{0, 1, 2, 3, 4, 5, 6\\}$ to the integers such that $f(0) = 0$, $f(6) = 12$, and\n",
      "\\[|x - y| \\leq |f(x) - f(y)| \\leq 3|x - y|\\]\n",
      "for all $x$ and $y$ in $\\{0, 1, 2, 3, 4, 5, 6\\}$.\n",
      "Found similar problem: Find the number of functions $f$ from $\\{0, 1, 2, 3, 4, 5, 6\\}$ to the integers such that $f(0) = 0$, $f(6) = 12$, and \\[|x - y| \\leq |f(x) - f(y)| \\leq 3|x - y|\\] for all $x$ and $y$ in $\\{0, 1, 2, 3, 4, 5, 6\\}$.\n",
      "\n",
      "1.0000001192092896\n",
      "The number $n$ can be written in base $14$ as $\\underline{a}\\text{ }\\underline{b}\\text{ }\\underline{c}$, can be written in base $15$ as $\\underline{a}\\text{ }\\underline{c}\\text{ }\\underline{b}$, and can be written in base $6$ as $\\underline{a}\\text{ }\\underline{c}\\text{ }\\underline{a}\\text{ }\\underline{c}\\text{ }$, where $a > 0$. Find the base-$10$ representation of $n$.\n",
      "Found similar problem: The number $n$ can be written in base $14$ as $\\underline{a}\\text{ }\\underline{b}\\text{ }\\underline{c}$, can be written in base $15$ as $\\underline{a}\\text{ }\\underline{c}\\text{ }\\underline{b}$, and can be written in base $6$ as $\\underline{a}\\text{ }\\underline{c}\\text{ }\\underline{a}\\text{ }\\underline{c}\\text{ }$, where $a > 0$. Find the base-$10$ representation of $n$.\n",
      "\n",
      "1.0\n",
      "Let $\\mathcal{S}$ be the set of all perfect squares whose rightmost three digits in base $10$ are $256$. Let $\\mathcal{T}$ be the set of all numbers of the form $\\frac{x-256}{1000}$, where $x$ is in $\\mathcal{S}$. In other words, $\\mathcal{T}$ is the set of numbers that result when the last three digits of each number in $\\mathcal{S}$ are truncated. Find the remainder when the tenth smallest element of $\\mathcal{T}$ is divided by $1000$.\n",
      "Found similar problem: Let $\\mathcal{S}$ be the set of all perfect squares whose rightmost three digits in base $10$ are $256$. Let $\\mathcal{T}$ be the set of all numbers of the form $\\frac{x-256}{1000}$, where $x$ is in $\\mathcal{S}$. In other words, $\\mathcal{T}$ is the set of numbers that result when the last three digits of each number in $\\mathcal{S}$ are truncated. Find the remainder when the tenth smallest element of $\\mathcal{T}$ is divided by $1000$.\n",
      "\n",
      "0.5039058923721313\n",
      "Al, Bert, and Carl are the winners of a school drawing for a pile of Halloween candy, which they are to divide in a ratio of $3:2:1$, respectively. Due to some confusion they come at different times to claim their prizes, and each assumes he is the first to arrive. If each takes what he believes to be the correct share of candy, what fraction of the candy goes unclaimed?\n",
      "$\\mathrm{(A) \\ } \\frac{1}{18}\\qquad \\mathrm{(B) \\ } \\frac{1}{6}\\qquad \\mathrm{(C) \\ } \\frac{2}{9}\\qquad \\mathrm{(D) \\ } \\frac{5}{18}\\qquad \\mathrm{(E) \\ } \\frac{5}{12}$\n",
      "Found similar problem: A game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from $$ 1$ to $$ 9999$ inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order A, B, C. As a hint, the digits of the three prices are given. On a particular day, the digits given were $1, 1, 1, 1, 3, 3, 3$. Find the total number of possible guesses for all three prizes consistent with the hint.\n",
      "\n",
      "1.0000001192092896\n",
      "Find the number of pairs $(m,n)$ of positive integers with $1\\le m<n\\le 30$ such that there exists a real number $x$ satisfying \\[\\sin(mx)+\\sin(nx)=2.\\]\n",
      "Found similar problem: Find the number of pairs $(m,n)$ of positive integers with $1\\le m<n\\le 30$ such that there exists a real number $x$ satisfying \\[\\sin(mx)+\\sin(nx)=2.\\]\n",
      "\n",
      "0.9999999403953552\n",
      "For every $m \\geq 2$, let $Q(m)$ be the least positive integer with the following property: For every $n \\geq Q(m)$, there is always a perfect cube $k^3$ in the range $n < k^3 \\leq m \\cdot n$. Find the remainder when \\[\\sum_{m = 2}^{2017} Q(m)\\]is divided by 1000.\n",
      "Found similar problem: For every $m \\geq 2$, let $Q(m)$ be the least positive integer with the following property: For every $n \\geq Q(m)$, there is always a perfect cube $k^3$ in the range $n < k^3 \\leq m \\cdot n$. Find the remainder when \\[\\sum_{m = 2}^{2017} Q(m)\\]is divided by 1000.\n",
      "\n",
      "0.9059549570083618\n",
      "A transformation of the first [quadrant](https://artofproblemsolving.com/wiki/index.php/Quadrant) of the [coordinate plane](https://artofproblemsolving.com/wiki/index.php/Coordinate_plane) maps each point $(x,y)$ to the point $(\\sqrt{x},\\sqrt{y}).$  The [vertices](https://artofproblemsolving.com/wiki/index.php/Vertex) of [quadrilateral](https://artofproblemsolving.com/wiki/index.php/Quadrilateral) $ABCD$ are $A=(900,300), B=(1800,600), C=(600,1800),$ and $D=(300,900).$  Let $k_{}$ be the area of the region enclosed by the image of quadrilateral $ABCD.$  Find the greatest integer that does not exceed $k_{}.$\n",
      "Found similar problem: A transformation of the first quadrant of the coordinate plane maps each point $(x,y)$ to the point $(\\sqrt{x},\\sqrt{y}).$ The vertices of quadrilateral $ABCD$ are $A=(900,300), B=(1800,600), C=(600,1800),$ and $D=(300,900).$ Let $k_{}$ be the area of the region enclosed by the image of quadrilateral $ABCD.$ Find the greatest integer that does not exceed $k_{}.$\n",
      "\n",
      "0.9999999403953552\n",
      "Find the number of $7$-tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following systems of equations:\n",
      "\\begin{align*} abc&=70,\\\\ cde&=71,\\\\ efg&=72. \\end{align*}\n",
      "Found similar problem: Find the number of $7$-tuples of positive integers $(a,b,c,d,e,f,g)$ that satisfy the following systems of equations: \\begin{align*} abc&=70,\\\\ cde&=71,\\\\ efg&=72. \\end{align*}\n",
      "\n",
      "0.9623492956161499\n",
      "Find the number of ordered triples $(a,b,c)$ where $a$, $b$, and $c$ are positive [integers](https://artofproblemsolving.com/wiki/index.php/Integer), $a$ is a [factor](https://artofproblemsolving.com/wiki/index.php/Factor) of $b$, $a$ is a factor of $c$, and $a+b+c=100$.\n",
      "Found similar problem: Find the number of ordered triples $(a,b,c)$ where $a$, $b$, and $c$ are positive integers, $a$ is a factor of $b$, $a$ is a factor of $c$, and $a+b+c=100$.\n",
      "\n",
      "0.9528191685676575\n",
      "Let the [set](https://artofproblemsolving.com/wiki/index.php/Set) $\\mathcal{S} = \\{8, 5, 1, 13, 34, 3, 21, 2\\}.$ Susan makes a list as follows: for each two-element subset of $\\mathcal{S},$ she writes on her list the greater of the set's two elements. Find the sum of the numbers on the list.\n",
      "Found similar problem: Let the set $\\mathcal{S} = \\{8, 5, 1, 13, 34, 3, 21, 2\\}.$ Susan makes a list as follows: for each two-element subset of $\\mathcal{S},$ she writes on her list the greater of the set's two elements. Find the sum of the numbers on the list.\n",
      "\n",
      "0.9701479077339172\n",
      "Two [squares](https://artofproblemsolving.com/wiki/index.php/Square) of a $7\\times 7$ checkerboard are painted yellow, and the rest are painted green. Two color schemes are equivalent if one can be obtained from the other by applying a [rotation](https://artofproblemsolving.com/wiki/index.php/Rotation) in the plane board. How many inequivalent color schemes are possible?\n",
      "Found similar problem: Two squares of a $7\\times 7$ checkerboard are painted yellow, and the rest are painted green. Two color schemes are equivalent if one can be obtained from the other by applying a rotation in the plane board. How many inequivalent color schemes are possible?\n",
      "\n",
      "0.9066290855407715\n",
      "The wheel shown below consists of two circles and five spokes, with a label at each point where a spoke meets a circle. A bug walks along the wheel, starting at point $A$. At every step of the process, the bug walks from one labeled point to an adjacent labeled point. Along the inner circle the bug only walks in a counterclockwise direction, and along the outer circle the bug only walks in a clockwise direction. For example, the bug could travel along the path $AJABCHCHIJA$, which has $10$ steps. Let $n$ be the number of paths with $15$ steps that begin and end at point $A$. Find the remainder when $n$ is divided by $1000.$\n",
      "Found similar problem: The wheel shown below consists of two circles and five spokes, with a label at each point where a spoke meets a circle. A bug walks along the wheel, starting at point $A$. At every step of the process, the bug walks from one labeled point to an adjacent labeled point. Along the inner circle the bug only walks in a counterclockwise direction, and along the outer circle the bug only walks in a clockwise direction. For example, the bug could travel along the path $AJABCHCHIJA$, which has $10$ steps. Let $n$ be the number of paths with $15$ steps that begin and end at point $A$. Find the remainder when $n$ is divided by $1000.$\n",
      "[asy] unitsize(32); draw(unitcircle); draw(scale(2) * unitcircle); for(int d = 90; d < 360 + 90; d += 72){ draw(2 * dir(d) -- dir(d)); } real s = 4; dot(1 * dir( 90), linewidth(s)); dot(1 * dir(162), linewidth(s)); dot(1 * dir(234), linewidth(s)); dot(1 * dir(306), linewidth(s)); dot(1 * dir(378), linewidth(s)); dot(2 * dir(378), linewidth(s)); dot(2 * dir(306), linewidth(s)); dot(2 * dir(234), linewidth(s)); dot(2 * dir(162), linewidth(s)); dot(2 * dir( 90), linewidth(s)); defaultpen(fontsize(10pt)); real r = 0.05; label(\"$A$\", (1-r) * dir( 90), -dir( 90)); label(\"$B$\", (1-r) * dir(162), -dir(162)); label(\"$C$\", (1-r) * dir(234), -dir(234)); label(\"$D$\", (1-r) * dir(306), -dir(306)); label(\"$E$\", (1-r) * dir(378), -dir(378)); label(\"$F$\", (2+r) * dir(378), dir(378)); label(\"$G$\", (2+r) * dir(306), dir(306)); label(\"$H$\", (2+r) * dir(234), dir(234)); label(\"$I$\", (2+r) * dir(162), dir(162)); label(\"$J$\", (2+r) * dir( 90), dir( 90)); [/asy]\n",
      "\n",
      "1.0\n",
      "A right prism with height $h$ has bases that are regular hexagons with sides of length $12$. A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and the face of the pyramid that does not contain $A$ measures $60$ degrees. Find $h^2$.\n",
      "Found similar problem: A right prism with height $h$ has bases that are regular hexagons with sides of length $12$. A vertex $A$ of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and the face of the pyramid that does not contain $A$ measures $60$ degrees. Find $h^2$.\n",
      "\n",
      "1.0\n",
      "The vertices of $\\triangle ABC$ are $A = (0,0)\\,$, $B = (0,420)\\,$, and $C = (560,0)\\,$.  The six faces of a die are labeled with two $A\\,$'s, two $B\\,$'s, and two $C\\,$'s.  Point $P_1 = (k,m)\\,$ is chosen in the interior of $\\triangle ABC$, and points $P_2\\,$, $P_3\\,$, $P_4, \\dots$ are generated by rolling the die repeatedly and applying the rule: If the die shows label $L\\,$, where $L \\in \\{A, B, C\\}$, and $P_n\\,$ is the most recently obtained point, then $P_{n + 1}^{}$ is the midpoint of $\\overline{P_n L}$.  Given that $P_7 = (14,92)\\,$, what is $k + m\\,$?\n",
      "Found similar problem: The vertices of $\\triangle ABC$ are $A = (0,0)\\,$, $B = (0,420)\\,$, and $C = (560,0)\\,$. The six faces of a die are labeled with two $A\\,$'s, two $B\\,$'s, and two $C\\,$'s. Point $P_1 = (k,m)\\,$ is chosen in the interior of $\\triangle ABC$, and points $P_2\\,$, $P_3\\,$, $P_4, \\dots$ are generated by rolling the die repeatedly and applying the rule: If the die shows label $L\\,$, where $L \\in \\{A, B, C\\}$, and $P_n\\,$ is the most recently obtained point, then $P_{n + 1}^{}$ is the midpoint of $\\overline{P_n L}$. Given that $P_7 = (14,92)\\,$, what is $k + m\\,$?\n",
      "\n",
      "1.0\n",
      "Let $x$, $y$ and $z$ all exceed $1$ and let $w$ be a positive number such that $\\log_x w = 24$, $\\log_y w = 40$ and $\\log_{xyz} w = 12$. Find $\\log_z w$.\n",
      "Found similar problem: Let $x$, $y$ and $z$ all exceed $1$ and let $w$ be a positive number such that $\\log_x w = 24$, $\\log_y w = 40$ and $\\log_{xyz} w = 12$. Find $\\log_z w$.\n",
      "\n",
      "1.0000001192092896\n",
      "For $t = 1, 2, 3, 4$, define $S_t = \\sum_{i = 1}^{350}a_i^t$, where $a_i \\in \\{1,2,3,4\\}$. If $S_1 = 513$ and $S_4 = 4745$, find the minimum possible value for $S_2$.\n",
      "Found similar problem: For $t = 1, 2, 3, 4$, define $S_t = \\sum_{i = 1}^{350}a_i^t$, where $a_i \\in \\{1,2,3,4\\}$. If $S_1 = 513$ and $S_4 = 4745$, find the minimum possible value for $S_2$.\n",
      "\n",
      "1.0\n",
      "Find the number of ordered pairs $(m, n)$ such that $m$ and $n$ are positive integers in the set $\\{1, 2, ..., 30\\}$ and the greatest common divisor of $2^m + 1$ and $2^n - 1$ is not $1$.\n",
      "Found similar problem: Find the number of ordered pairs $(m, n)$ such that $m$ and $n$ are positive integers in the set $\\{1, 2, ..., 30\\}$ and the greatest common divisor of $2^m + 1$ and $2^n - 1$ is not $1$.\n",
      "\n",
      "1.0\n",
      "Suppose that the measurement of time during the day is converted to the metric system so that each day has $10$ metric hours, and each metric hour has $100$ metric minutes. Digital clocks would then be produced that would read $\\text{9:99}$ just before midnight, $\\text{0:00}$ at midnight, $\\text{1:25}$ at the former $\\text{3:00}$ AM, and $\\text{7:50}$ at the former $\\text{6:00}$ PM. After the conversion, a person who wanted to wake up at the equivalent of the former $\\text{6:36}$ AM would set his new digital alarm clock for $\\text{A:BC}$, where $\\text{A}$, $\\text{B}$, and $\\text{C}$ are digits. Find $100\\text{A}+10\\text{B}+\\text{C}$.\n",
      "Found similar problem: Suppose that the measurement of time during the day is converted to the metric system so that each day has $10$ metric hours, and each metric hour has $100$ metric minutes. Digital clocks would then be produced that would read $\\text{9:99}$ just before midnight, $\\text{0:00}$ at midnight, $\\text{1:25}$ at the former $\\text{3:00}$ AM, and $\\text{7:50}$ at the former $\\text{6:00}$ PM. After the conversion, a person who wanted to wake up at the equivalent of the former $\\text{6:36}$ AM would set his new digital alarm clock for $\\text{A:BC}$, where $\\text{A}$, $\\text{B}$, and $\\text{C}$ are digits. Find $100\\text{A}+10\\text{B}+\\text{C}$.\n",
      "\n",
      "0.9110072255134583\n",
      "[Rectangle](https://artofproblemsolving.com/wiki/index.php/Rectangle) $ABCD$ is given with $AB=63$ and $BC=448.$ Points $E$ and $F$ lie on $AD$ and $BC$ respectively, such that $AE=CF=84.$ The [inscribed circle](https://artofproblemsolving.com/wiki/index.php/Inscribed_circle) of [triangle](https://artofproblemsolving.com/wiki/index.php/Triangle) $BEF$ is [tangent](https://artofproblemsolving.com/wiki/index.php/Tangent) to $EF$ at point $P,$ and the inscribed circle of triangle $DEF$ is tangent to $EF$ at [point](https://artofproblemsolving.com/wiki/index.php/Point) $Q.$ Find $PQ.$\n",
      "Found similar problem: Rectangle $ABCD$ is given with $AB=63$ and $BC=448.$ Points $E$ and $F$ lie on $AD$ and $BC$ respectively, such that $AE=CF=84.$ The inscribed circle of triangle $BEF$ is tangent to $EF$ at point $P,$ and the inscribed circle of triangle $DEF$ is tangent to $EF$ at point $Q.$ Find $PQ.$\n",
      "\n",
      "1.0000001192092896\n",
      "In equiangular octagon $CAROLINE$, $CA = RO = LI = NE =$ $\\sqrt{2}$ and $AR = OL = IN = EC = 1$. The self-intersecting octagon $CORNELIA$ encloses six non-overlapping triangular regions. Let $K$ be the area enclosed by $CORNELIA$, that is, the total area of the six triangular regions. Then $K =$ $\\dfrac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.\n",
      "Found similar problem: In equiangular octagon $CAROLINE$, $CA = RO = LI = NE =$ $\\sqrt{2}$ and $AR = OL = IN = EC = 1$. The self-intersecting octagon $CORNELIA$ encloses six non-overlapping triangular regions. Let $K$ be the area enclosed by $CORNELIA$, that is, the total area of the six triangular regions. Then $K =$ $\\dfrac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a + b$.\n",
      "\n",
      "1.0\n",
      "Two positive integers differ by $60$. The sum of their square roots is the square root of an integer that is not a perfect square. What is the maximum possible sum of the two integers?\n",
      "Found similar problem: Two positive integers differ by $60$. The sum of their square roots is the square root of an integer that is not a perfect square. What is the maximum possible sum of the two integers?\n",
      "\n",
      "1.0\n",
      "The table below displays some of the results of last summer's Frostbite Falls Fishing Festival, showing how many contestants caught $n\\,$ fish for various values of $n\\,$. \n",
      "\n",
      "$\\begin{array}{|c|c|c|c|c|c|c|c|c|} \\hline n & 0 & 1 & 2 & 3 & \\dots & 13 & 14 & 15 \\\\ \\hline \\text{number of contestants who caught} \\ n \\ \\text{fish} & 9 & 5 & 7 & 23 & \\dots & 5 & 2 & 1 \\\\ \\hline \\end{array}$\n",
      "In the newspaper story covering the event, it was reported that \n",
      "\n",
      "(a) the winner caught $15$ fish;\n",
      "(b) those who caught $3$ or more fish averaged $6$ fish each;\n",
      "(c) those who caught $12$ or fewer fish averaged $5$ fish each.\n",
      "What was the total number of fish caught during the festival?\n",
      "Found similar problem: The table below displays some of the results of last summer's Frostbite Falls Fishing Festival, showing how many contestants caught $n\\,$ fish for various values of $n\\,$.\n",
      "$\\begin{array}{|c|c|c|c|c|c|c|c|c|} \\hline n & 0 & 1 & 2 & 3 & \\dots & 13 & 14 & 15 \\\\ \\hline \\text{number of contestants who caught} \\ n \\ \\text{fish} & 9 & 5 & 7 & 23 & \\dots & 5 & 2 & 1 \\\\ \\hline \\end{array}$\n",
      " In the newspaper story covering the event, it was reported that\n",
      " (a) the winner caught $15$ fish;\n",
      " (b) those who caught $3$ or more fish averaged $6$ fish each;\n",
      " (c) those who caught $12$ or fewer fish averaged $5$ fish each.\n",
      " What was the total number of fish caught during the festival?\n",
      "\n",
      "0.952359676361084\n",
      "Two thousand points are given on a [circle](https://artofproblemsolving.com/wiki/index.php/Circle). Label one of the points $1$. From this point, count $2$ points in the clockwise direction and label this point $2$. From the point labeled $2$, count $3$ points in the clockwise direction and label this point $3$. (See figure.) Continue this process until the labels $1,2,3\\dots,1993\\,$ are all used. Some of the points on the circle will have more than one label and some points will not have a label. What is the smallest integer that labels the same point as $1993$? \n",
      "[AIME 1993 Problem 9.png](https://artofproblemsolving.com/wiki/index.php/File:AIME_1993_Problem_9.png)\n",
      "Found similar problem: Two thousand points are given on a circle. Label one of the points $1$. From this point, count $2$ points in the clockwise direction and label this point $2$. From the point labeled $2$, count $3$ points in the clockwise direction and label this point $3$. (See figure.) Continue this process until the labels $1,2,3\\dots,1993\\,$ are all used. Some of the points on the circle will have more than one label and some points will not have a label. What is the smallest integer that labels the same point as $1993$?\n",
      "\n",
      "1.0\n",
      "Find the number of positive integers $m$ for which there exist nonnegative integers $x_0$, $x_1$ , $\\dots$ , $x_{2011}$ such that\n",
      "\\[m^{x_0} = \\sum_{k = 1}^{2011} m^{x_k}.\\]\n",
      "Found similar problem: Find the number of positive integers $m$ for which there exist nonnegative integers $x_0$, $x_1$ , $\\dots$ , $x_{2011}$ such that \\[m^{x_0} = \\sum_{k = 1}^{2011} m^{x_k}.\\]\n",
      "\n",
      "0.9386293888092041\n",
      "An [integer](https://artofproblemsolving.com/wiki/index.php/Integer) is called parity-monotonic if its decimal representation $a_{1}a_{2}a_{3}\\cdots a_{k}$ satisfies $a_{i}<a_{i+1}$ if $a_{i}$ is [odd](https://artofproblemsolving.com/wiki/index.php/Odd), and $a_{i}>a_{i+1}$ if $a_{i}$ is [even](https://artofproblemsolving.com/wiki/index.php/Even). How many four-digit parity-monotonic integers are there?\n",
      "Found similar problem: An integer is called parity-monotonic if its decimal representation $a_{1}a_{2}a_{3}\\cdots a_{k}$ satisfies $a_{i}<a_{i+1}$ if $a_{i}$ is odd, and $a_{i}>a_{i+1}$ if $a_{i}$ is even. How many four-digit parity-monotonic integers are there?\n",
      "\n",
      "1.0000001192092896\n",
      "A club consisting of $11$ men and $12$ women needs to choose a committee from among its members so that the number of women on the committee is one more than the number of men on the committee. The committee could have as few as $1$ member or as many as $23$ members. Let $N$ be the number of such committees that can be formed. Find the sum of the prime numbers that divide $N.$\n",
      "Found similar problem: A club consisting of $11$ men and $12$ women needs to choose a committee from among its members so that the number of women on the committee is one more than the number of men on the committee. The committee could have as few as $1$ member or as many as $23$ members. Let $N$ be the number of such committees that can be formed. Find the sum of the prime numbers that divide $N.$\n",
      "\n",
      "0.8783740401268005\n",
      "Squares $S_1$ and $S_2$ are [inscribed](https://artofproblemsolving.com/wiki/index.php/Inscribe) in [right triangle](https://artofproblemsolving.com/wiki/index.php/Right_triangle) $ABC$, as shown in the figures below. Find $AC + CB$ if area $(S_1) = 441$ and area $(S_2) = 440$.\n",
      "[AIME 1987 Problem 15.png](https://artofproblemsolving.com/wiki/index.php/File:AIME_1987_Problem_15.png)\n",
      "Found similar problem: Squares $S_1$ and $S_2$ are inscribed in right triangle $ABC$, as shown in the figures below. Find $AC + CB$ if area $(S_1) = 441$ and area $(S_2) = 440$.\n",
      "\n",
      "1.0\n",
      "A circle is circumscribed around an isosceles triangle whose two congruent angles have degree measure $x$. Two points are chosen independently and uniformly at random on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is $\\frac{14}{25}$. Find the difference between the largest and smallest possible values of $x$.\n",
      "Found similar problem: A circle is circumscribed around an isosceles triangle whose two congruent angles have degree measure $x$. Two points are chosen independently and uniformly at random on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is $\\frac{14}{25}$. Find the difference between the largest and smallest possible values of $x$.\n",
      "\n",
      "0.9957438707351685\n",
      "David found four sticks of different lengths that can be used to form three non-congruent convex cyclic quadrilaterals, $A,\\text{ }B,\\text{ }C$, which can each be inscribed in a circle with radius $1$. Let $\\varphi_A$ denote the measure of the acute angle made by the diagonals of quadrilateral $A$, and define $\\varphi_B$ and $\\varphi_C$ similarly. Suppose that $\\sin\\varphi_A=\\tfrac{2}{3}$, $\\sin\\varphi_B=\\tfrac{3}{5}$, and $\\sin\\varphi_C=\\tfrac{6}{7}$. All three quadrilaterals have the same area $K$, which can be written in the form $\\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n",
      "Found similar problem: David found four sticks of different lengths that can be used to form three non-congruent convex cyclic quadrilaterals, $A,\\text{ }B,\\text{ }C$, which can each be inscribed in a circle with radius $1$. Let $\\varphi_A$ denote the measure of the acute angle made by the diagonals of quadrilateral $A$, and define $\\varphi_B$ and $\\varphi_C$ similarly. Suppose that $\\sin\\varphi_A=\\frac{2}{3}$, $\\sin\\varphi_B=\\frac{3}{5}$, and $\\sin\\varphi_C=\\frac{6}{7}$. All three quadrilaterals have the same area $K$, which can be written in the form $\\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n",
      "\n",
      "1.0\n",
      "Triangle $ABC_0$ has a right angle at $C_0$. Its side lengths are pairwise relatively prime positive integers, and its perimeter is $p$. Let $C_1$ be the foot of the altitude to $\\overline{AB}$, and for $n \\geq 2$, let $C_n$ be the foot of the altitude to $\\overline{C_{n-2}B}$ in $\\triangle C_{n-2}C_{n-1}B$. The sum $\\sum_{n=2}^\\infty C_{n-2}C_{n-1} = 6p$. Find $p$.\n",
      "Found similar problem: Triangle $ABC_0$ has a right angle at $C_0$. Its side lengths are pairwise relatively prime positive integers, and its perimeter is $p$. Let $C_1$ be the foot of the altitude to $\\overline{AB}$, and for $n \\geq 2$, let $C_n$ be the foot of the altitude to $\\overline{C_{n-2}B}$ in $\\triangle C_{n-2}C_{n-1}B$. The sum $\\sum_{n=2}^\\infty C_{n-2}C_{n-1} = 6p$. Find $p$.\n",
      "\n",
      "1.0\n",
      "A solitaire game is played as follows.  Six distinct pairs of matched tiles are placed in a bag.  The player randomly draws tiles one at a time from the bag and retains them, except that matching tiles are put aside as soon as they appear in the player's hand.  The game ends if the player ever holds three tiles, no two of which match; otherwise the drawing continues until the bag is empty.  The probability that the bag will be emptied is $p/q,\\,$ where $p\\,$ and $q\\,$ are relatively prime positive integers.  Find $p+q.\\,$\n",
      "Found similar problem: A solitaire game is played as follows. Six distinct pairs of matched tiles are placed in a bag. The player randomly draws tiles one at a time from the bag and retains them, except that matching tiles are put aside as soon as they appear in the player's hand. The game ends if the player ever holds three tiles, no two of which match; otherwise the drawing continues until the bag is empty. The probability that the bag will be emptied is $p/q,\\,$ where $p\\,$ and $q\\,$ are relatively prime positive integers. Find $p+q.\\,$\n",
      "\n",
      "1.0\n",
      "Define a domino to be an ordered pair of distinct positive integers. A proper sequence of dominos is a list of distinct dominos in which the first coordinate of each pair after the first equals the second coordinate of the immediately preceding pair, and in which $(i,j)$ and $(j,i)$ do not both appear for any $i$ and $j$.  Let $D_{40}$ be the set of all dominos whose coordinates are no larger than 40.  Find the length of the longest proper sequence of dominos that can be formed using the dominos of $D_{40}.$\n",
      "Found similar problem: Define a domino to be an ordered pair of distinct positive integers. A proper sequence of dominos is a list of distinct dominos in which the first coordinate of each pair after the first equals the second coordinate of the immediately preceding pair, and in which $(i,j)$ and $(j,i)$ do not both appear for any $i$ and $j$. Let $D_{40}$ be the set of all dominos whose coordinates are no larger than 40. Find the length of the longest proper sequence of dominos that can be formed using the dominos of $D_{40}.$\n",
      "\n",
      "0.980629026889801\n",
      "In a 6 x 4 grid (6 rows, 4 columns), 12 of the 24 squares are to be shaded so that there are two shaded squares in each row and three shaded squares in each column.  Let $N$ be the number of shadings with this property.  Find the remainder when $N$ is divided by 1000.\n",
      "[AIME I 2007-10.png](https://artofproblemsolving.com/wiki/index.php/File:AIME_I_2007-10.png)\n",
      "Found similar problem: In a 6 x 4 grid (6 rows, 4 columns), 12 of the 24 squares are to be shaded so that there are two shaded squares in each row and three shaded squares in each column. Let $N$ be the number of shadings with this property. Find the remainder when $N$ is divided by 1000.\n",
      "\n",
      "1.0\n",
      "The system of equations\n",
      "\\begin{eqnarray*}\\log_{10}(2000xy) - (\\log_{10}x)(\\log_{10}y) & = & 4 \\\\ \\log_{10}(2yz) - (\\log_{10}y)(\\log_{10}z) & = & 1 \\\\ \\log_{10}(zx) - (\\log_{10}z)(\\log_{10}x) & = & 0 \\\\ \\end{eqnarray*}\n",
      "has two solutions $(x_{1},y_{1},z_{1})$ and $(x_{2},y_{2},z_{2})$. Find $y_{1} + y_{2}$.\n",
      "Found similar problem: The system of equations \\begin{eqnarray*}\\log_{10}(2000xy) - (\\log_{10}x)(\\log_{10}y) & = & 4 \\\\ \\log_{10}(2yz) - (\\log_{10}y)(\\log_{10}z) & = & 1 \\\\ \\log_{10}(zx) - (\\log_{10}z)(\\log_{10}x) & = & 0 \\\\ \\end{eqnarray*}\n",
      " has two solutions $(x_{1},y_{1},z_{1})$ and $(x_{2},y_{2},z_{2})$. Find $y_{1} + y_{2}$.\n",
      "\n",
      "1.0\n",
      "Find the number of four-element subsets of $\\{1,2,3,4,\\dots, 20\\}$ with the property that two distinct elements of a subset have a sum of $16$, and two distinct elements of a subset have a sum of $24$. For example, $\\{3,5,13,19\\}$ and $\\{6,10,20,18\\}$ are two such subsets.\n",
      "Found similar problem: Find the number of four-element subsets of $\\{1,2,3,4,\\dots, 20\\}$ with the property that two distinct elements of a subset have a sum of $16$, and two distinct elements of a subset have a sum of $24$. For example, $\\{3,5,13,19\\}$ and $\\{6,10,20,18\\}$ are two such subsets.\n",
      "\n",
      "1.0\n",
      "Find the least positive integer $m$ such that $m^2 - m + 11$ is a product of at least four not necessarily distinct primes.\n",
      "Found similar problem: Find the least positive integer $m$ such that $m^2 - m + 11$ is a product of at least four not necessarily distinct primes.\n",
      "\n",
      "0.9054882526397705\n",
      "Suppose $n$ is a [positive integer](https://artofproblemsolving.com/wiki/index.php/Positive_integer) and $d$ is a single [digit](https://artofproblemsolving.com/wiki/index.php/Digit) in [base 10](https://artofproblemsolving.com/wiki/index.php/Base_10). Find $n$ if\n",
      "\n",
      "$\\frac{n}{810}=0.d25d25d25\\ldots$\n",
      "Found similar problem: Suppose $n$ is a positive integer and $d$ is a single digit in base 10. Find $n$ if\n",
      "$\\frac{n}{810}=0.d25d25d25\\ldots$\n",
      "\n",
      "0.9456037282943726\n",
      "[Point](https://artofproblemsolving.com/wiki/index.php/Point) $B$ is in the exterior of the [regular](https://artofproblemsolving.com/wiki/index.php/Regular_polygon) $n$-sided polygon $A_1A_2\\cdots A_n$, and $A_1A_2B$ is an [equilateral triangle](https://artofproblemsolving.com/wiki/index.php/Equilateral_triangle). What is the largest value of $n$ for which $A_1$, $A_n$, and $B$ are consecutive vertices of a regular polygon?\n",
      "Found similar problem: Point $B$ is in the exterior of the regular $n$-sided polygon $A_1A_2\\cdots A_n$, and $A_1A_2B$ is an equilateral triangle. What is the largest value of $n$ for which $A_1$, $A_n$, and $B$ are consecutive vertices of a regular polygon?\n",
      "\n",
      "1.0000001192092896\n",
      "A group of clerks is assigned the task of sorting $1775$ files. Each clerk sorts at a constant rate of $30$ files per hour. At the end of the first hour, some of the clerks are reassigned to another task; at the end of the second hour, the same number of the remaining clerks are also reassigned to another task, and a similar assignment occurs at the end of the third hour. The group finishes the sorting in $3$ hours and $10$ minutes. Find the number of files sorted during the first one and a half hours of sorting.\n",
      "Found similar problem: A group of clerks is assigned the task of sorting $1775$ files. Each clerk sorts at a constant rate of $30$ files per hour. At the end of the first hour, some of the clerks are reassigned to another task; at the end of the second hour, the same number of the remaining clerks are also reassigned to another task, and a similar assignment occurs at the end of the third hour. The group finishes the sorting in $3$ hours and $10$ minutes. Find the number of files sorted during the first one and a half hours of sorting.\n",
      "\n",
      "1.0000001192092896\n",
      "Find the number of ordered pairs of positive integers $(m,n)$ such that ${m^2n = 20 ^{20}}$.\n",
      "Found similar problem: Find the number of ordered pairs of positive integers $(m,n)$ such that ${m^2n = 20 ^{20}}$.\n",
      "\n",
      "1.0\n",
      "A basketball player has a constant probability of $.4$ of making any given shot, independent of previous shots. Let $a_n$ be the ratio of shots made to shots attempted after $n$ shots. The probability that $a_{10} = .4$ and $a_n\\le.4$ for all $n$ such that $1\\le n\\le9$ is given to be $p^aq^br/\\left(s^c\\right)$ where $p$, $q$, $r$, and $s$ are primes, and $a$, $b$, and $c$ are positive integers. Find $\\left(p+q+r+s\\right)\\left(a+b+c\\right)$.\n",
      "Found similar problem: A basketball player has a constant probability of $.4$ of making any given shot, independent of previous shots. Let $a_n$ be the ratio of shots made to shots attempted after $n$ shots. The probability that $a_{10} = .4$ and $a_n\\le.4$ for all $n$ such that $1\\le n\\le9$ is given to be $p^aq^br/\\left(s^c\\right)$ where $p$, $q$, $r$, and $s$ are primes, and $a$, $b$, and $c$ are positive integers. Find $\\left(p+q+r+s\\right)\\left(a+b+c\\right)$.\n",
      "\n",
      "0.9303703904151917\n",
      "For nonnegative integers $a$ and $b$ with  $a + b \\leq 6$, let $T(a, b) = \\binom{6}{a} \\binom{6}{b} \\binom{6}{a + b}$. Let $S$ denote the sum of all $T(a, b)$, where  $a$ and $b$ are nonnegative integers with $a + b \\leq 6$. Find the remainder when $S$ is divided by $1000$.\n",
      "Found similar problem: For nonnegative integers $a$ and $b$ with $a + b \\leq 6$, let $T(a, b) = \\binom{6}{a} \\binom{6}{b} \\binom{6}{a + b}$. Let $S$ denote the sum of all $T(a, b)$, where $a$ and $b$ are nonnegative integers with $a + b \\leq 6$. Find the remainder when $S$ is divided by $1000$.\n",
      "Major Note\n",
      " Most solutions use committee forming (except for the bash solution). To understand more about the techniques used, visit the committee forming page for more information.\n",
      "\n",
      "1.0\n",
      "Let $x$ be a real number such that $\\sin^{10}x+\\cos^{10} x = \\tfrac{11}{36}$. Then $\\sin^{12}x+\\cos^{12} x = \\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n",
      "Found similar problem: Let $x$ be a real number such that $\\sin^{10}x+\\cos^{10} x = \\tfrac{11}{36}$. Then $\\sin^{12}x+\\cos^{12} x = \\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n",
      "\n",
      "1.0\n",
      "Call a permutation $a_1, a_2, \\ldots, a_n$ of the integers $1, 2, \\ldots, n$ quasi-increasing if $a_k \\leq a_{k+1} + 2$ for each $1 \\leq k \\leq n-1$. For example, 53421 and 14253 are quasi-increasing permutations of the integers $1, 2, 3, 4, 5$, but 45123 is not. Find the number of quasi-increasing permutations of the integers $1, 2, \\ldots, 7$.\n",
      "Found similar problem: Call a permutation $a_1, a_2, \\ldots, a_n$ of the integers $1, 2, \\ldots, n$ quasi-increasing if $a_k \\leq a_{k+1} + 2$ for each $1 \\leq k \\leq n-1$. For example, 53421 and 14253 are quasi-increasing permutations of the integers $1, 2, 3, 4, 5$, but 45123 is not. Find the number of quasi-increasing permutations of the integers $1, 2, \\ldots, 7$.\n",
      "\n",
      "1.0\n",
      "Jon and Steve ride their bicycles along a path that parallels two side-by-side train tracks running the east/west direction. Jon rides east at $20$ miles per hour, and Steve rides west at $20$ miles per hour. Two trains of equal length, traveling in opposite directions at constant but different speeds each pass the two riders. Each train takes exactly $1$ minute to go past Jon. The westbound train takes $10$ times as long as the eastbound train to go past Steve. The length of each train is $\\tfrac{m}{n}$ miles, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n",
      "Found similar problem: Jon and Steve ride their bicycles along a path that parallels two side-by-side train tracks running the east/west direction. Jon rides east at $20$ miles per hour, and Steve rides west at $20$ miles per hour. Two trains of equal length, traveling in opposite directions at constant but different speeds each pass the two riders. Each train takes exactly $1$ minute to go past Jon. The westbound train takes $10$ times as long as the eastbound train to go past Steve. The length of each train is $\\tfrac{m}{n}$ miles, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n",
      "\n",
      "1.0\n",
      "Two unit squares are selected at random without replacement from an $n \\times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected unit squares are horizontally or vertically adjacent is less than $\\frac{1}{2015}$.\n",
      "Found similar problem: Two unit squares are selected at random without replacement from an $n \\times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected unit squares are horizontally or vertically adjacent is less than $\\frac{1}{2015}$.\n",
      "\n",
      "0.9999998807907104\n",
      "In a drawer Sandy has $5$ pairs of socks, each pair a different color.  On Monday Sandy selects two individual socks at random from the $10$ socks in the drawer.  On Tuesday Sandy selects $2$ of the remaining $8$ socks at random and on Wednesday two of the remaining $6$ socks at random.  The probability that Wednesday is the first day Sandy selects matching socks is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers,  Find $m+n$.\n",
      "Found similar problem: In a drawer Sandy has $5$ pairs of socks, each pair a different color. On Monday Sandy selects two individual socks at random from the $10$ socks in the drawer. On Tuesday Sandy selects $2$ of the remaining $8$ socks at random and on Wednesday two of the remaining $6$ socks at random. The probability that Wednesday is the first day Sandy selects matching socks is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, Find $m+n$.\n",
      "\n",
      "0.9374860525131226\n",
      "Determine the number of [ordered pairs](https://artofproblemsolving.com/wiki/index.php/Ordered_pair) $(a,b)$ of [integers](https://artofproblemsolving.com/wiki/index.php/Integer) such that $\\log_a b + 6\\log_b a=5, 2 \\leq a \\leq 2005,$ and $2 \\leq b \\leq 2005.$\n",
      "Found similar problem: Determine the number of ordered pairs $(a,b)$ of integers such that $\\log_a b + 6\\log_b a=5, 2 \\leq a \\leq 2005,$ and $2 \\leq b \\leq 2005.$\n",
      "\n",
      "0.9675877690315247\n",
      "[Set](https://artofproblemsolving.com/wiki/index.php/Set) $A$ consists of $m$ consecutive integers whose sum is $2m$, and set $B$ consists of $2m$ consecutive integers whose sum is $m.$ The absolute value of the difference between the greatest element of $A$ and the greatest element of $B$ is $99$. Find $m.$\n",
      "Found similar problem: Set $A$ consists of $m$ consecutive integers whose sum is $2m$, and set $B$ consists of $2m$ consecutive integers whose sum is $m.$ The absolute value of the difference between the greatest element of $A$ and the greatest element of $B$ is $99$. Find $m.$\n",
      "\n",
      "1.0000001192092896\n",
      "Find the number of positive integers less than $1000$ that can be expressed as the difference of two integral powers of $2.$\n",
      "Found similar problem: Find the number of positive integers less than $1000$ that can be expressed as the difference of two integral powers of $2.$\n",
      "\n",
      "1.0\n",
      "Define a sequence recursively by $t_1 = 20$, $t_2 = 21$, and\\[t_n = \\frac{5t_{n-1}+1}{25t_{n-2}}\\]for all $n \\ge 3$. Then $t_{2020}$ can be expressed as $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.\n",
      "Found similar problem: Define a sequence recursively by $t_1 = 20$, $t_2 = 21$, and\\[t_n = \\frac{5t_{n-1}+1}{25t_{n-2}}\\]for all $n \\ge 3$. Then $t_{2020}$ can be expressed as $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.\n",
      "\n",
      "0.8277010917663574\n",
      "Three 12 cm $\\times$12 cm [ squares](https://artofproblemsolving.com/wiki/index.php/Square_(geometry)) are each cut into two pieces $A$ and $B$, as shown in the first figure below, by joining the [midpoints](https://artofproblemsolving.com/wiki/index.php/Midpoint) of two adjacent sides. These six pieces are then attached to a [ regular](https://artofproblemsolving.com/wiki/index.php/Regular_polygon) [hexagon](https://artofproblemsolving.com/wiki/index.php/Hexagon), as shown in the second figure, so as to fold into a [polyhedron](https://artofproblemsolving.com/wiki/index.php/Polyhedron). What is the [volume](https://artofproblemsolving.com/wiki/index.php/Volume) (in $\\mathrm{cm}^3$) of this polyhedron?\n",
      "[AIME 1985 Problem 15.png](https://artofproblemsolving.com/wiki/index.php/File:AIME_1985_Problem_15.png)\n",
      "Found similar problem: Three 12 cm $\\times$12 cm squares are each cut into two pieces $A$ and $B$, as shown in the first figure below, by joining the midpoints of two adjacent sides. These six pieces are then attached to a regular hexagon, as shown in the second figure, so as to fold into a polyhedron. What is the volume (in $\\mathrm{cm}^3$) of this polyhedron?\n",
      "\n",
      "1.0000001192092896\n",
      "Find the number of integers $c$ such that the equation \\[\\left||20|x|-x^2|-c\\right|=21\\]has $12$ distinct real solutions.\n",
      "Found similar problem: Find the number of integers $c$ such that the equation \\[\\left||20|x|-x^2|-c\\right|=21\\]has $12$ distinct real solutions.\n",
      "\n",
      "1.0\n",
      "Before starting to paint, Bill had $130$ ounces of blue paint, $164$ ounces of red paint, and $188$ ounces of white paint. Bill painted four equally sized stripes on a wall, making a blue stripe, a red stripe, a white stripe, and a pink stripe. Pink is a mixture of red and white, not necessarily in equal amounts. When Bill finished, he had equal amounts of blue, red, and white paint left. Find the total number of ounces of paint Bill had left.\n",
      "Found similar problem: Before starting to paint, Bill had $130$ ounces of blue paint, $164$ ounces of red paint, and $188$ ounces of white paint. Bill painted four equally sized stripes on a wall, making a blue stripe, a red stripe, a white stripe, and a pink stripe. Pink is a mixture of red and white, not necessarily in equal amounts. When Bill finished, he had equal amounts of blue, red, and white paint left. Find the total number of ounces of paint Bill had left.\n",
      "\n",
      "1.0000001192092896\n",
      "The value of $x$ that satisfies $\\log_{2^x} 3^{20} = \\log_{2^{x+3}} 3^{2020}$ can be written as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n",
      "Found similar problem: The value of $x$ that satisfies $\\log_{2^x} 3^{20} = \\log_{2^{x+3}} 3^{2020}$ can be written as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n",
      "\n",
      "1.000000238418579\n",
      "A hexagon that is inscribed in a circle has side lengths $22$, $22$, $20$, $22$, $22$, and $20$ in that order. The radius of the circle can be written as $p+\\sqrt{q}$, where $p$ and $q$ are positive integers. Find $p+q$.\n",
      "Found similar problem: A hexagon that is inscribed in a circle has side lengths $22$, $22$, $20$, $22$, $22$, and $20$ in that order. The radius of the circle can be written as $p+\\sqrt{q}$, where $p$ and $q$ are positive integers. Find $p+q$.\n",
      "\n",
      "1.0000001192092896\n",
      "A group of children held a grape-eating contest. When the contest was over, the winner had eaten $n$ grapes, and the child in $k$-th place had eaten  $n+2-2k$ grapes. The total number of grapes eaten in the contest was $2009$. Find the smallest possible value of $n$.\n",
      "Found similar problem: A group of children held a grape-eating contest. When the contest was over, the winner had eaten $n$ grapes, and the child in $k$-th place had eaten $n+2-2k$ grapes. The total number of grapes eaten in the contest was $2009$. Find the smallest possible value of $n$.\n",
      "\n",
      "1.0000001192092896\n",
      "Let $f(x)=(x^2+3x+2)^{\\cos(\\pi x)}$. Find the sum of all positive integers $n$ for which \n",
      "\\[\\left |\\sum_{k=1}^n\\log_{10}f(k)\\right|=1.\\]\n",
      "Found similar problem: Let $f(x)=(x^2+3x+2)^{\\cos(\\pi x)}$. Find the sum of all positive integers $n$ for which \\[\\left |\\sum_{k=1}^n\\log_{10}f(k)\\right|=1.\\]\n",
      "\n",
      "1.0\n",
      "Let $A_1,A_2,A_3,\\cdots,A_{12}$ be the vertices of a regular dodecagon. How many distinct squares in the plane of the dodecagon have at least two vertices in the set $\\{A_1,A_2,A_3,\\cdots,A_{12}\\} ?$\n",
      "Found similar problem: Let $A_1,A_2,A_3,\\cdots,A_{12}$ be the vertices of a regular dodecagon. How many distinct squares in the plane of the dodecagon have at least two vertices in the set $\\{A_1,A_2,A_3,\\cdots,A_{12}\\} ?$\n",
      "\n",
      "1.0\n",
      "Square $AIME$ has sides of length $10$ units.  Isosceles triangle $GEM$ has base $EM$, and the area common to triangle $GEM$ and square $AIME$ is $80$ square units.  Find the length of the altitude to $EM$ in $\\triangle GEM$.\n",
      "Found similar problem: Square $AIME$ has sides of length $10$ units. Isosceles triangle $GEM$ has base $EM$, and the area common to triangle $GEM$ and square $AIME$ is $80$ square units. Find the length of the altitude to $EM$ in $\\triangle GEM$.\n",
      "\n",
      "1.0\n",
      "Anh read a book. On the first day she read $n$ pages in $t$ minutes, where $n$ and $t$ are positive integers. On the second day Anh read $n + 1$ pages in $t + 1$ minutes. Each day thereafter Anh read one more page than she read on the previous day, and it took her one more minute than on the previous day until she completely read the $374$ page book. It took her a total of $319$ minutes to read the book. Find $n + t$.\n",
      "Found similar problem: Anh read a book. On the first day she read $n$ pages in $t$ minutes, where $n$ and $t$ are positive integers. On the second day Anh read $n + 1$ pages in $t + 1$ minutes. Each day thereafter Anh read one more page than she read on the previous day, and it took her one more minute than on the previous day until she completely read the $374$ page book. It took her a total of $319$ minutes to read the book. Find $n + t$.\n",
      "\n",
      "1.0\n",
      "A pyramid has a triangular base with side lengths $20$, $20$, and $24$. The three edges of the pyramid from the three corners of the base to the fourth vertex of the pyramid all have length $25$. The volume of the pyramid is $m\\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.\n",
      "Found similar problem: A pyramid has a triangular base with side lengths $20$, $20$, and $24$. The three edges of the pyramid from the three corners of the base to the fourth vertex of the pyramid all have length $25$. The volume of the pyramid is $m\\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.\n",
      "\n",
      "1.0\n",
      "Kathy has $5$ red cards and $5$ green cards. She shuffles the $10$ cards and lays out $5$ of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders RRGGG, GGGGR, or RRRRR will make Kathy happy, but RRRGR will not. The probability that Kathy will be happy is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.\n",
      "Found similar problem: Kathy has $5$ red cards and $5$ green cards. She shuffles the $10$ cards and lays out $5$ of the cards in a row in a random order. She will be happy if and only if all the red cards laid out are adjacent and all the green cards laid out are adjacent. For example, card orders RRGGG, GGGGR, or RRRRR will make Kathy happy, but RRRGR will not. The probability that Kathy will be happy is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.\n",
      "\n",
      "1.0\n",
      "Initially Alex, Betty, and Charlie had a total of $444$ peanuts. Charlie had the most peanuts, and Alex had the least. The three numbers of peanuts that each person had formed a geometric progression. Alex eats $5$ of his peanuts, Betty eats $9$ of her peanuts, and Charlie eats $25$ of his peanuts. Now the three numbers of peanuts each person has forms an arithmetic progression. Find the number of peanuts Alex had initially.\n",
      "Found similar problem: Initially Alex, Betty, and Charlie had a total of $444$ peanuts. Charlie had the most peanuts, and Alex had the least. The three numbers of peanuts that each person had formed a geometric progression. Alex eats $5$ of his peanuts, Betty eats $9$ of her peanuts, and Charlie eats $25$ of his peanuts. Now the three numbers of peanuts each person has forms an arithmetic progression. Find the number of peanuts Alex had initially.\n",
      "\n",
      "0.9589849710464478\n",
      "Three planets orbit a star circularly in the same plane.  Each moves in the same direction and moves at [constant](https://artofproblemsolving.com/wiki/index.php/Constant) speed.  Their periods are 60, 84, and 140 years.  The three planets and the star are currently [collinear](https://artofproblemsolving.com/wiki/index.php/Collinear).  What is the fewest number of years from now that they will all be collinear again?\n",
      "Found similar problem: Three planets orbit a star circularly in the same plane. Each moves in the same direction and moves at constant speed. Their periods are 60, 84, and 140 years. The three planets and the star are currently collinear. What is the fewest number of years from now that they will all be collinear again?\n",
      "\n",
      "0.9687255620956421\n",
      "Every positive [integer](https://artofproblemsolving.com/wiki/index.php/Integer) $k$ has a unique factorial base expansion $(f_1,f_2,f_3,\\ldots,f_m)$, meaning that $k=1!\\cdot f_1+2!\\cdot f_2+3!\\cdot f_3+\\cdots+m!\\cdot f_m$, where each $f_i$ is an integer, $0\\le f_i\\le i$, and $0<f_m$. Given that $(f_1,f_2,f_3,\\ldots,f_j)$ is the factorial base expansion of $16!-32!+48!-64!+\\cdots+1968!-1984!+2000!$, find the value of $f_1-f_2+f_3-f_4+\\cdots+(-1)^{j+1}f_j$.\n",
      "Found similar problem: Every positive integer $k$ has a unique factorial base expansion $(f_1,f_2,f_3,\\ldots,f_m)$, meaning that $k=1!\\cdot f_1+2!\\cdot f_2+3!\\cdot f_3+\\cdots+m!\\cdot f_m$, where each $f_i$ is an integer, $0\\le f_i\\le i$, and $0<f_m$. Given that $(f_1,f_2,f_3,\\ldots,f_j)$ is the factorial base expansion of $16!-32!+48!-64!+\\cdots+1968!-1984!+2000!$, find the value of $f_1-f_2+f_3-f_4+\\cdots+(-1)^{j+1}f_j$.\n",
      "\n",
      "1.0\n",
      "Find the least positive integer $n$ such that no matter how $10^{n}$ is expressed as the product of any two positive integers, at least one of these two integers contains the digit $0$.\n",
      "Found similar problem: Find the least positive integer $n$ such that no matter how $10^{n}$ is expressed as the product of any two positive integers, at least one of these two integers contains the digit $0$.\n",
      "\n",
      "1.0\n",
      "Find the number of ordered pairs of positive integer solutions $(m, n)$ to the equation $20m + 12n = 2012$.\n",
      "Found similar problem: Find the number of ordered pairs of positive integer solutions $(m, n)$ to the equation $20m + 12n = 2012$.\n",
      "\n",
      "0.9550981521606445\n",
      "Let $(a,b,c)$ be the [real](https://artofproblemsolving.com/wiki/index.php/Real_number) solution of the system of equations $x^3 - xyz = 2$, $y^3 - xyz = 6$, $z^3 - xyz = 20$. The greatest possible value of $a^3 + b^3 + c^3$ can be written in the form $\\frac {m}{n}$, where $m$ and $n$ are [relatively prime](https://artofproblemsolving.com/wiki/index.php/Relatively_prime) positive integers. Find $m + n$.\n",
      "Found similar problem: Let $(a,b,c)$ be the real solution of the system of equations $x^3 - xyz = 2$, $y^3 - xyz = 6$, $z^3 - xyz = 20$. The greatest possible value of $a^3 + b^3 + c^3$ can be written in the form $\\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.\n",
      "\n",
      "0.9217179417610168\n",
      "The diagram shows twenty congruent [circles](https://artofproblemsolving.com/wiki/index.php/Circle) arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the rectangle as shown in the diagram. The [ratio](https://artofproblemsolving.com/wiki/index.php/Ratio) of the longer dimension of the rectangle to the shorter dimension can be written as $\\dfrac{1}{2}(\\sqrt{p}-q)$ where $p$ and $q$ are positive integers. Find $p+q$.\n",
      "\n",
      "[AIME 2002I Problem 02.png](https://artofproblemsolving.com/wiki/index.php/File:AIME_2002I_Problem_02.png)\n",
      "Found similar problem: The diagram shows twenty congruent circles arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the rectangle as shown in the diagram. The ratio of the longer dimension of the rectangle to the shorter dimension can be written as $\\dfrac{1}{2}(\\sqrt{p}-q)$ where $p$ and $q$ are positive integers. Find $p+q$.\n",
      "\n",
      "0.23341092467308044\n",
      "In how many years, approximately, from 1998 will the population of Nisos be as much as Queen Irene has proclaimed that the islands can support?\n",
      "$\\text{(A)}\\ 50\\text{ yrs.} \\qquad \\text{(B)}\\ 75\\text{ yrs.} \\qquad \\text{(C)}\\ 100\\text{ yrs.} \\qquad \\text{(D)}\\ 125\\text{ yrs.} \\qquad \\text{(E)}\\ 150\\text{ yrs.}$\n",
      "Found similar problem: A biologist wants to calculate the number of fish in a lake. On May 1 she catches a random sample of 60 fish, tags them, and releases them. On September 1 she catches a random sample of 70 fish and finds that 3 of them are tagged. To calculate the number of fish in the lake on May 1, she assumes that 25% of these fish are no longer in the lake on September 1 (because of death and emigrations), that 40% of the fish were not in the lake May 1 (because of births and immigrations), and that the number of untagged fish and tagged fish in the September 1 sample are representative of the total population. What does the biologist calculate for the number of fish in the lake on May 1?\n",
      "\n",
      "0.9391507506370544\n",
      "A sample of 121 [integers](https://artofproblemsolving.com/wiki/index.php/Integer) is given, each between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique [mode](https://artofproblemsolving.com/wiki/index.php/Mode) (most frequent value). Let $D$ be the difference between the mode and the [arithmetic mean](https://artofproblemsolving.com/wiki/index.php/Arithmetic_mean) of the sample. What is the largest possible value of $\\lfloor D\\rfloor$? (For real $x$, $\\lfloor x\\rfloor$ is the [greatest integer](https://artofproblemsolving.com/wiki/index.php/Floor_function) less than or equal to $x$.)\n",
      "Found similar problem: A sample of 121 integers is given, each between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique mode (most frequent value). Let $D$ be the difference between the mode and the arithmetic mean of the sample. What is the largest possible value of $\\lfloor D\\rfloor$? (For real $x$, $\\lfloor x\\rfloor$ is the greatest integer less than or equal to $x$.)\n",
      "\n",
      "1.0\n",
      "A frog is positioned at the origin of the coordinate plane. From the point $(x, y)$, the frog can jump to any of the points $(x + 1, y)$, $(x + 2, y)$, $(x, y + 1)$, or $(x, y + 2)$. Find the number of distinct sequences of jumps in which the frog begins at $(0, 0)$ and ends at $(4, 4)$.\n",
      "Found similar problem: A frog is positioned at the origin of the coordinate plane. From the point $(x, y)$, the frog can jump to any of the points $(x + 1, y)$, $(x + 2, y)$, $(x, y + 1)$, or $(x, y + 2)$. Find the number of distinct sequences of jumps in which the frog begins at $(0, 0)$ and ends at $(4, 4)$.\n",
      "\n",
      "1.0000001192092896\n",
      "Find the number of positive integers with three not necessarily distinct digits, $abc$, with $a \\neq 0$ and $c \\neq 0$ such that both $abc$ and $cba$ are multiples of $4$.\n",
      "Found similar problem: Find the number of positive integers with three not necessarily distinct digits, $abc$, with $a \\neq 0$ and $c \\neq 0$ such that both $abc$ and $cba$ are multiples of $4$.\n",
      "\n",
      "1.0\n",
      "For positive integers $n$ and $k$, let $f(n, k)$ be the remainder when $n$ is divided by $k$, and for $n > 1$ let $F(n) = \\max_{\\substack{1\\le k\\le \\frac{n}{2}}} f(n, k)$. Find the remainder when $\\sum\\limits_{n=20}^{100} F(n)$ is divided by $1000$.\n",
      "Found similar problem: For positive integers $n$ and $k$, let $f(n, k)$ be the remainder when $n$ is divided by $k$, and for $n > 1$ let $F(n) = \\max_{\\substack{1\\le k\\le \\frac{n}{2}}} f(n, k)$. Find the remainder when $\\sum\\limits_{n=20}^{100} F(n)$ is divided by $1000$.\n",
      "\n",
      "0.969031810760498\n",
      "One commercially available ten-button lock may be opened by pressing -- in any order -- the correct five buttons. The sample shown below has $\\{1,2,3,6,9\\}$ as its [combination](https://artofproblemsolving.com/wiki/index.php/Combination). Suppose that these locks are redesigned so that sets of as many as nine buttons or as few as one button could serve as combinations. How many additional combinations would this allow?\n",
      "[1988-1.png](https://artofproblemsolving.com/wiki/index.php/File:1988-1.png)\n",
      "Found similar problem: One commercially available ten-button lock may be opened by pressing -- in any order -- the correct five buttons. The sample shown below has $\\{1,2,3,6,9\\}$ as its combination. Suppose that these locks are redesigned so that sets of as many as nine buttons or as few as one button could serve as combinations. How many additional combinations would this allow?\n",
      "\n",
      "1.0\n",
      "Let $P(x) = x^2 - 3x - 7$, and let $Q(x)$ and $R(x)$ be two quadratic polynomials also with the coefficient of $x^2$ equal to $1$. David computes each of the three sums $P + Q$, $P + R$, and $Q + R$ and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If $Q(0) = 2$, then $R(0) = \\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.\n",
      "Found similar problem: Let $P(x) = x^2 - 3x - 7$, and let $Q(x)$ and $R(x)$ be two quadratic polynomials also with the coefficient of $x^2$ equal to $1$. David computes each of the three sums $P + Q$, $P + R$, and $Q + R$ and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If $Q(0) = 2$, then $R(0) = \\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.\n",
      "\n",
      "1.0000001192092896\n",
      "Jenn randomly chooses a number $J$ from $1, 2, 3,\\ldots, 19, 20$. Bela then randomly chooses a number $B$ from $1, 2, 3,\\ldots, 19, 20$ distinct from $J$. The value of $B - J$ is at least $2$ with a probability that can be expressed in the form $\\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n",
      "Found similar problem: Jenn randomly chooses a number $J$ from $1, 2, 3,\\ldots, 19, 20$. Bela then randomly chooses a number $B$ from $1, 2, 3,\\ldots, 19, 20$ distinct from $J$. The value of $B - J$ is at least $2$ with a probability that can be expressed in the form $\\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n",
      "\n",
      "1.0\n",
      "Beatrix is going to place six rooks on a $6 \\times 6$ chessboard where both the rows and columns are labeled $1$ to $6$; the rooks are placed so that no two rooks are in the same row or the same column. The $value$ of a square is the sum of its row number and column number. The $score$ of an arrangement of rooks is the least value of any occupied square.The average score over all valid configurations is $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.\n",
      "Found similar problem: Beatrix is going to place six rooks on a $6 \\times 6$ chessboard where both the rows and columns are labeled $1$ to $6$; the rooks are placed so that no two rooks are in the same row or the same column. The $value$ of a square is the sum of its row number and column number. The $score$ of an arrangement of rooks is the least value of any occupied square.The average score over all valid configurations is $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.\n",
      "\n",
      "1.0\n",
      "For any finite set $S$, let $|S|$ denote the number of elements in $S$. Find the number of ordered pairs $(A,B)$ such that $A$ and $B$ are (not necessarily distinct) subsets of $\\{1,2,3,4,5\\}$ that satisfy \\[|A| \\cdot |B| = |A \\cap B| \\cdot |A \\cup B|\\]\n",
      "Found similar problem: For any finite set $S$, let $|S|$ denote the number of elements in $S$. Find the number of ordered pairs $(A,B)$ such that $A$ and $B$ are (not necessarily distinct) subsets of $\\{1,2,3,4,5\\}$ that satisfy \\[|A| \\cdot |B| = |A \\cap B| \\cdot |A \\cup B|\\]\n",
      "\n",
      "0.963866114616394\n",
      "Rudolph bikes at a [constant](https://artofproblemsolving.com/wiki/index.php/Constant) rate and stops for a five-minute break at the end of every mile. Jennifer bikes at a constant rate which is three-quarters the rate that Rudolph bikes, but Jennifer takes a five-minute break at the end of every two miles. Jennifer and Rudolph begin biking at the same time and arrive at the $50$-mile mark at exactly the same time. How many minutes has it taken them?\n",
      "Found similar problem: Rudolph bikes at a constant rate and stops for a five-minute break at the end of every mile. Jennifer bikes at a constant rate which is three-quarters the rate that Rudolph bikes, but Jennifer takes a five-minute break at the end of every two miles. Jennifer and Rudolph begin biking at the same time and arrive at the $50$-mile mark at exactly the same time. How many minutes has it taken them?\n",
      "\n",
      "1.0000001192092896\n",
      "Al walks down to the bottom of an escalator that is moving up and he counts 150 steps. His friend, Bob, walks up to the top of the escalator and counts 75 steps. If Al's speed of walking (in steps per unit time) is three times Bob's walking speed, how many steps are visible on the escalator at a given time? (Assume that this value is constant.)\n",
      "Found similar problem: Al walks down to the bottom of an escalator that is moving up and he counts 150 steps. His friend, Bob, walks up to the top of the escalator and counts 75 steps. If Al's speed of walking (in steps per unit time) is three times Bob's walking speed, how many steps are visible on the escalator at a given time? (Assume that this value is constant.)\n",
      "\n",
      "1.0\n",
      "Points $A$, $B$, and $C$ lie in that order along a straight path where the distance from $A$ to $C$ is $1800$ meters. Ina runs twice as fast as Eve, and Paul runs twice as fast as Ina. The three runners start running at the same time with Ina starting at $A$ and running toward $C$, Paul starting at $B$ and running toward $C$, and Eve starting at $C$ and running toward $A$. When Paul meets Eve, he turns around and runs toward $A$. Paul and Ina both arrive at $B$ at the same time. Find the number of meters from $A$ to $B$.\n",
      "Found similar problem: Points $A$, $B$, and $C$ lie in that order along a straight path where the distance from $A$ to $C$ is $1800$ meters. Ina runs twice as fast as Eve, and Paul runs twice as fast as Ina. The three runners start running at the same time with Ina starting at $A$ and running toward $C$, Paul starting at $B$ and running toward $C$, and Eve starting at $C$ and running toward $A$. When Paul meets Eve, he turns around and runs toward $A$. Paul and Ina both arrive at $B$ at the same time. Find the number of meters from $A$ to $B$.\n",
      "\n",
      "1.0\n",
      "For any positive integer $k$, let $f_1(k)$ denote the square of the sum of the digits of $k$.  For $n \\ge 2$, let $f_n(k) = f_1(f_{n - 1}(k))$.  Find $f_{1988}(11)$.\n",
      "Found similar problem: For any positive integer $k$, let $f_1(k)$ denote the square of the sum of the digits of $k$. For $n \\ge 2$, let $f_n(k) = f_1(f_{n - 1}(k))$. Find $f_{1988}(11)$.\n",
      "\n",
      "1.0\n",
      "Find the sum of all positive integers $b < 1000$ such that the base-$b$ integer $36_{b}$ is a perfect square and the base-$b$ integer $27_{b}$ is a perfect cube.\n",
      "Found similar problem: Find the sum of all positive integers $b < 1000$ such that the base-$b$ integer $36_{b}$ is a perfect square and the base-$b$ integer $27_{b}$ is a perfect cube.\n",
      "\n",
      "1.0\n",
      "Fifteen distinct points are designated on $\\triangle ABC$: the 3 vertices $A$, $B$, and $C$; $3$ other points on side $\\overline{AB}$; $4$ other points on side $\\overline{BC}$; and $5$ other points on side $\\overline{CA}$. Find the number of triangles with positive area whose vertices are among these $15$ points.\n",
      "Found similar problem: Fifteen distinct points are designated on $\\triangle ABC$: the 3 vertices $A$, $B$, and $C$; $3$ other points on side $\\overline{AB}$; $4$ other points on side $\\overline{BC}$; and $5$ other points on side $\\overline{CA}$. Find the number of triangles with positive area whose vertices are among these $15$ points.\n",
      "\n",
      "1.0000001192092896\n",
      "Let $f(n)$ and $g(n)$ be functions satisfying\n",
      "\\[f(n) = \\begin{cases}\\sqrt{n} & \\text{ if } \\sqrt{n} \\text{ is an integer}\\\\ 1 + f(n+1) & \\text{ otherwise} \\end{cases}\\]and\n",
      "\\[g(n) = \\begin{cases}\\sqrt{n} & \\text{ if } \\sqrt{n} \\text{ is an integer}\\\\ 2 + g(n+2) & \\text{ otherwise} \\end{cases}\\]for positive integers $n$. Find the least positive integer $n$ such that $\\tfrac{f(n)}{g(n)} = \\tfrac{4}{7}$.\n",
      "Found similar problem: Let $f(n)$ and $g(n)$ be functions satisfying \\[f(n) = \\begin{cases}\\sqrt{n} & \\text{ if } \\sqrt{n} \\text{ is an integer}\\\\ 1 + f(n+1) & \\text{ otherwise} \\end{cases}\\]and \\[g(n) = \\begin{cases}\\sqrt{n} & \\text{ if } \\sqrt{n} \\text{ is an integer}\\\\ 2 + g(n+2) & \\text{ otherwise} \\end{cases}\\]for positive integers $n$. Find the least positive integer $n$ such that $\\tfrac{f(n)}{g(n)} = \\tfrac{4}{7}$.\n",
      "\n",
      "1.0\n",
      "Suppose that $y = \\frac34x$ and $x^y = y^x$. The quantity $x + y$ can be expressed as a rational number $\\frac {r}{s}$, where $r$ and $s$ are relatively prime positive integers. Find $r + s$.\n",
      "Found similar problem: Suppose that $y = \\frac34x$ and $x^y = y^x$. The quantity $x + y$ can be expressed as a rational number $\\frac {r}{s}$, where $r$ and $s$ are relatively prime positive integers. Find $r + s$.\n",
      "\n",
      "1.0\n",
      "There are $2^{10} = 1024$ possible 10-letter strings in which each letter is either an A or a B. Find the number of such strings that do not have more than 3 adjacent letters that are identical.\n",
      "Found similar problem: There are $2^{10} = 1024$ possible 10-letter strings in which each letter is either an A or a B. Find the number of such strings that do not have more than 3 adjacent letters that are identical.\n",
      "\n",
      "0.9999999403953552\n",
      "For certain pairs $(m,n)$ of positive integers with $m\\geq n$ there are exactly $50$ distinct positive integers $k$ such that $|\\log m - \\log k| < \\log n$. Find the sum of all possible values of the product $mn$.\n",
      "Found similar problem: For certain pairs $(m,n)$ of positive integers with $m\\geq n$ there are exactly $50$ distinct positive integers $k$ such that $|\\log m - \\log k| < \\log n$. Find the sum of all possible values of the product $mn$.\n",
      "\n",
      "1.0000001192092896\n",
      "Call a set $S$ product-free if there do not exist $a, b, c \\in S$ (not necessarily distinct) such that $a b = c$. For example, the empty set and the set $\\{16, 20\\}$ are product-free, whereas the sets $\\{4, 16\\}$ and $\\{2, 8, 16\\}$ are not product-free. Find the number of product-free subsets of the set $\\{1, 2, 3, 4,..., 7, 8, 9, 10\\}$.\n",
      "Found similar problem: Call a set $S$ product-free if there do not exist $a, b, c \\in S$ (not necessarily distinct) such that $a b = c$. For example, the empty set and the set $\\{16, 20\\}$ are product-free, whereas the sets $\\{4, 16\\}$ and $\\{2, 8, 16\\}$ are not product-free. Find the number of product-free subsets of the set $\\{1, 2, 3, 4,..., 7, 8, 9, 10\\}$.\n",
      "\n",
      "1.0\n",
      "Find the number of positive integers $n$ less than $2017$ such that \\[1+n+\\frac{n^2}{2!}+\\frac{n^3}{3!}+\\frac{n^4}{4!}+\\frac{n^5}{5!}+\\frac{n^6}{6!}\\] is an integer.\n",
      "Found similar problem: Find the number of positive integers $n$ less than $2017$ such that \\[1+n+\\frac{n^2}{2!}+\\frac{n^3}{3!}+\\frac{n^4}{4!}+\\frac{n^5}{5!}+\\frac{n^6}{6!}\\] is an integer.\n",
      "\n",
      "1.0\n",
      "There exist unique positive integers $x$ and $y$ that satisfy the equation $x^2 + 84x + 2008 = y^2$. Find $x + y$.\n",
      "Found similar problem: There exist unique positive integers $x$ and $y$ that satisfy the equation $x^2 + 84x + 2008 = y^2$. Find $x + y$.\n",
      "\n",
      "1.0\n",
      "Let the set $S = \\{P_1, P_2, \\dots, P_{12}\\}$ consist of the twelve vertices of a regular $12$-gon. A subset $Q$ of $S$ is called \"communal\" if there is a circle such that all points of $Q$ are inside the circle, and all points of $S$ not in $Q$ are outside of the circle. How many communal subsets are there? (Note that the empty set is a communal subset.)\n",
      "Found similar problem: Let the set $S = \\{P_1, P_2, \\dots, P_{12}\\}$ consist of the twelve vertices of a regular $12$-gon. A subset $Q$ of $S$ is called \"communal\" if there is a circle such that all points of $Q$ are inside the circle, and all points of $S$ not in $Q$ are outside of the circle. How many communal subsets are there? (Note that the empty set is a communal subset.)\n",
      "\n",
      "0.9286762475967407\n",
      "Let $C$ be the [graph](https://artofproblemsolving.com/wiki/index.php/Graph) of $xy = 1$, and denote by $C^*$ the [reflection](https://artofproblemsolving.com/wiki/index.php/Reflection) of $C$ in the line $y = 2x$.  Let the [equation](https://artofproblemsolving.com/wiki/index.php/Equation) of $C^*$ be written in the form\n",
      "\\[12x^2 + bxy + cy^2 + d = 0.\\]\n",
      "Find the product $bc$.\n",
      "Found similar problem: Let $C$ be the graph of $xy = 1$, and denote by $C^*$ the reflection of $C$ in the line $y = 2x$. Let the equation of $C^*$ be written in the form\n",
      "\\[12x^2 + bxy + cy^2 + d = 0.\\]\n",
      " Find the product $bc$.\n",
      "\n",
      "1.0\n",
      "Triangle $ABC$ has side lengths $AB = 9$, $BC =$ $5\\sqrt{3}$, and $AC = 12$. Points $A = P_{0}, P_{1}, P_{2}, ... , P_{2450} = B$ are on segment $\\overline{AB}$ with $P_{k}$ between $P_{k-1}$ and $P_{k+1}$ for $k = 1, 2, ..., 2449$, and points $A = Q_{0}, Q_{1}, Q_{2}, ... , Q_{2450} = C$ are on segment $\\overline{AC}$ with $Q_{k}$ between $Q_{k-1}$ and $Q_{k+1}$ for $k = 1, 2, ..., 2449$. Furthermore, each segment $\\overline{P_{k}Q_{k}}$, $k = 1, 2, ..., 2449$, is parallel to $\\overline{BC}$. The segments cut the triangle into $2450$ regions, consisting of $2449$ trapezoids and $1$ triangle. Each of the $2450$ regions has the same area. Find the number of segments $\\overline{P_{k}Q_{k}}$, $k = 1, 2, ..., 2450$, that have rational length.\n",
      "Found similar problem: Triangle $ABC$ has side lengths $AB = 9$, $BC =$ $5\\sqrt{3}$, and $AC = 12$. Points $A = P_{0}, P_{1}, P_{2}, ... , P_{2450} = B$ are on segment $\\overline{AB}$ with $P_{k}$ between $P_{k-1}$ and $P_{k+1}$ for $k = 1, 2, ..., 2449$, and points $A = Q_{0}, Q_{1}, Q_{2}, ... , Q_{2450} = C$ are on segment $\\overline{AC}$ with $Q_{k}$ between $Q_{k-1}$ and $Q_{k+1}$ for $k = 1, 2, ..., 2449$. Furthermore, each segment $\\overline{P_{k}Q_{k}}$, $k = 1, 2, ..., 2449$, is parallel to $\\overline{BC}$. The segments cut the triangle into $2450$ regions, consisting of $2449$ trapezoids and $1$ triangle. Each of the $2450$ regions has the same area. Find the number of segments $\\overline{P_{k}Q_{k}}$, $k = 1, 2, ..., 2450$, that have rational length.\n",
      "\n",
      "0.9568815231323242\n",
      "Find the number of [positive integers](https://artofproblemsolving.com/wiki/index.php/Positive_integer) that are divisors of at least one of $10^{10},15^7,18^{11}.$\n",
      "Found similar problem: Find the number of positive integers that are divisors of at least one of $10^{10},15^7,18^{11}.$\n",
      "\n",
      "0.9595516920089722\n",
      "Square $S_{1}$ is $1\\times 1.$  For $i\\ge 1,$ the lengths of the sides of square $S_{i+1}$ are half the lengths of the sides of square $S_{i},$ two adjacent sides of square $S_{i}$ are perpendicular bisectors of two adjacent sides of square $S_{i+1},$ and the other two sides of square $S_{i+1},$ are the perpendicular bisectors of two adjacent sides of square $S_{i+2}.$  The total area enclosed by at least one of $S_{1}, S_{2}, S_{3}, S_{4}, S_{5}$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers.  Find $m-n.$\n",
      "[AIME 1995 Problem 1.png](https://artofproblemsolving.com/wiki/index.php/File:AIME_1995_Problem_1.png)\n",
      "Found similar problem: Square $S_{1}$ is $1\\times 1.$ For $i\\ge 1,$ the lengths of the sides of square $S_{i+1}$ are half the lengths of the sides of square $S_{i},$ two adjacent sides of square $S_{i}$ are perpendicular bisectors of two adjacent sides of square $S_{i+1},$ and the other two sides of square $S_{i+1},$ are the perpendicular bisectors of two adjacent sides of square $S_{i+2}.$ The total area enclosed by at least one of $S_{1}, S_{2}, S_{3}, S_{4}, S_{5}$ can be written in the form $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m-n.$\n",
      "\n",
      "1.0\n",
      "Let $N$ be the number of ordered triples $(A,B,C)$ of integers satisfying the conditions \n",
      "(a) $0\\le A<B<C\\le99$, \n",
      "(b) there exist integers $a$, $b$, and $c$, and prime $p$ where $0\\le b<a<c<p$, \n",
      "(c) $p$ divides $A-a$, $B-b$, and $C-c$, and \n",
      "(d) each ordered triple $(A,B,C)$ and each ordered triple $(b,a,c)$ form arithmetic sequences. Find $N$.\n",
      "Found similar problem: Let $N$ be the number of ordered triples $(A,B,C)$ of integers satisfying the conditions (a) $0\\le A<B<C\\le99$, (b) there exist integers $a$, $b$, and $c$, and prime $p$ where $0\\le b<a<c<p$, (c) $p$ divides $A-a$, $B-b$, and $C-c$, and (d) each ordered triple $(A,B,C)$ and each ordered triple $(b,a,c)$ form arithmetic sequences. Find $N$.\n",
      "\n",
      "1.0\n",
      "In a new school $40$ percent of the students are freshmen, $30$ percent are sophomores, $20$ percent are juniors, and $10$ percent are seniors. All freshmen are required to take Latin, and $80$ percent of the sophomores, $50$ percent of the juniors, and $20$ percent of the seniors elect to take Latin. The probability that a randomly chosen Latin student is a sophomore is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n",
      "Found similar problem: In a new school $40$ percent of the students are freshmen, $30$ percent are sophomores, $20$ percent are juniors, and $10$ percent are seniors. All freshmen are required to take Latin, and $80$ percent of the sophomores, $50$ percent of the juniors, and $20$ percent of the seniors elect to take Latin. The probability that a randomly chosen Latin student is a sophomore is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n",
      "\n",
      "1.0\n",
      "Ana, Bob, and CAO bike at constant rates of $8.6$ meters per second, $6.2$ meters per second, and $5$ meters per second, respectively. They all begin biking at the same time from the northeast corner of a rectangular field whose longer side runs due west. Ana starts biking along the edge of the field, initially heading west, Bob starts biking along the edge of the field, initially heading south, and Cao bikes in a straight line across the field to a point $D$ on the south edge of the field. Cao arrives at point $D$ at the same time that Ana and Bob arrive at $D$ for the first time. The ratio of the field's length to the field's width to the distance from point $D$ to the southeast corner of the field can be represented as $p : q : r$, where $p$, $q$, and $r$ are positive integers with $p$ and $q$ relatively prime. Find $p+q+r$.\n",
      "Found similar problem: Ana, Bob, and Cao bike at constant rates of $8.6$ meters per second, $6.2$ meters per second, and $5$ meters per second, respectively. They all begin biking at the same time from the northeast corner of a rectangular field whose longer side runs due west. Ana starts biking along the edge of the field, initially heading west, Bob starts biking along the edge of the field, initially heading south, and Cao bikes in a straight line across the field to a point $D$ on the south edge of the field. Cao arrives at point $D$ at the same time that Ana and Bob arrive at $D$ for the first time. The ratio of the field's length to the field's width to the distance from point $D$ to the southeast corner of the field can be represented as $p : q : r$, where $p$, $q$, and $r$ are positive integers with $p$ and $q$ relatively prime. Find $p+q+r$.\n",
      "\n",
      "1.0\n",
      "For any positive integer $a, \\sigma(a)$ denotes the sum of the positive integer divisors of $a$. Let $n$ be the least positive integer such that $\\sigma(a^n)-1$ is divisible by $2021$ for all positive integers $a$. Find the sum of the prime factors in the prime factorization of $n$.\n",
      "Found similar problem: For any positive integer $a, \\sigma(a)$ denotes the sum of the positive integer divisors of $a$. Let $n$ be the least positive integer such that $\\sigma(a^n)-1$ is divisible by $2021$ for all positive integers $a$. Find the sum of the prime factors in the prime factorization of $n$.\n",
      "\n",
      "1.0\n",
      "Positive integers $a$ and $b$ satisfy the condition\n",
      "\\[\\log_2(\\log_{2^a}(\\log_{2^b}(2^{1000}))) = 0.\\]\n",
      "Find the sum of all possible values of $a+b$.\n",
      "Found similar problem: Positive integers $a$ and $b$ satisfy the condition \\[\\log_2(\\log_{2^a}(\\log_{2^b}(2^{1000}))) = 0.\\] Find the sum of all possible values of $a+b$.\n",
      "\n",
      "0.9055576324462891\n",
      "Equilateral triangle $ABC$ has side length $840$. Point $D$ lies on the same side of line $BC$ as $A$ such that $\\overline{BD} \\perp \\overline{BC}$. The line $\\ell$ through $D$ parallel to line $BC$ intersects sides $\\overline{AB}$ and $\\overline{AC}$ at points $E$ and $F$, respectively. Point $G$ lies on $\\ell$ such that $F$ is between $E$ and $G$, $\\triangle AFG$ is isosceles, and the ratio of the area of $\\triangle AFG$ to the area of $\\triangle BED$ is $8:9$. Find $AF$.\n",
      "Found similar problem: Equilateral triangle $ABC$ has side length $840$. Point $D$ lies on the same side of line $BC$ as $A$ such that $\\overline{BD} \\perp \\overline{BC}$. The line $\\ell$ through $D$ parallel to line $BC$ intersects sides $\\overline{AB}$ and $\\overline{AC}$ at points $E$ and $F$, respectively. Point $G$ lies on $\\ell$ such that $F$ is between $E$ and $G$, $\\triangle AFG$ is isosceles, and the ratio of the area of $\\triangle AFG$ to the area of $\\triangle BED$ is $8:9$. Find $AF$.\n",
      "Diagram\n",
      "[asy] pair A,B,C,D,E,F,G; B=origin; A=5*dir(60); C=(5,0); E=0.6*A+0.4*B; F=0.6*A+0.4*C; G=rotate(240,F)*A; D=extension(E,F,B,dir(90)); draw(D--G--A,grey); draw(B--0.5*A+rotate(60,B)*A*0.5,grey); draw(A--B--C--cycle,linewidth(1.5)); dot(A^^B^^C^^D^^E^^F^^G); label(\"$A$\",A,dir(90)); label(\"$B$\",B,dir(225)); label(\"$C$\",C,dir(-45)); label(\"$D$\",D,dir(180)); label(\"$E$\",E,dir(-45)); label(\"$F$\",F,dir(225)); label(\"$G$\",G,dir(0)); label(\"$\\ell$\",midpoint(E--F),dir(90)); [/asy]\n",
      "\n",
      "1.0\n",
      "A right hexagonal prism has height $2$. The bases are regular hexagons with side length $1$. Any $3$ of the $12$ vertices determine a triangle. Find the number of these triangles that are isosceles (including equilateral triangles).\n",
      "Found similar problem: A right hexagonal prism has height $2$. The bases are regular hexagons with side length $1$. Any $3$ of the $12$ vertices determine a triangle. Find the number of these triangles that are isosceles (including equilateral triangles).\n",
      "\n",
      "1.0\n",
      "In the array of 13 squares shown below, 8 squares are colored red, and the remaining 5 squares are colored blue. If one of all possible such colorings is chosen at random, the probability that the chosen colored array appears the same when rotated 90 degrees around the central square is $\\frac{1}{n}$ , where n is a positive integer. Find n.\n",
      "[asy] draw((0,0)--(1,0)--(1,1)--(0,1)--(0,0)); draw((2,0)--(2,2)--(3,2)--(3,0)--(3,1)--(2,1)--(4,1)--(4,0)--(2,0)); draw((1,2)--(1,4)--(0,4)--(0,2)--(0,3)--(1,3)--(-1,3)--(-1,2)--(1,2)); draw((-1,1)--(-3,1)--(-3,0)--(-1,0)--(-2,0)--(-2,1)--(-2,-1)--(-1,-1)--(-1,1)); draw((0,-1)--(0,-3)--(1,-3)--(1,-1)--(1,-2)--(0,-2)--(2,-2)--(2,-1)--(0,-1)); size(100);[/asy]\n",
      "Found similar problem: In the array of 13 squares shown below, 8 squares are colored red, and the remaining 5 squares are colored blue. If one of all possible such colorings is chosen at random, the probability that the chosen colored array appears the same when rotated 90 degrees around the central square is $\\frac{1}{n}$ , where n is a positive integer. Find n.\n",
      "[asy] draw((0,0)--(1,0)--(1,1)--(0,1)--(0,0)); draw((2,0)--(2,2)--(3,2)--(3,0)--(3,1)--(2,1)--(4,1)--(4,0)--(2,0)); draw((1,2)--(1,4)--(0,4)--(0,2)--(0,3)--(1,3)--(-1,3)--(-1,2)--(1,2)); draw((-1,1)--(-3,1)--(-3,0)--(-1,0)--(-2,0)--(-2,1)--(-2,-1)--(-1,-1)--(-1,1)); draw((0,-1)--(0,-3)--(1,-3)--(1,-1)--(1,-2)--(0,-2)--(2,-2)--(2,-1)--(0,-1)); size(100);[/asy]\n",
      "\n",
      "1.0000001192092896\n",
      "Let $m$ and $n$ be odd integers greater than $1.$ An $m\\times n$ rectangle is made up of unit squares where the squares in the top row are numbered left to right with the integers $1$ through $n$, those in the second row are numbered left to right with the integers $n + 1$ through $2n$, and so on. Square $200$ is in the top row, and square $2000$ is in the bottom row. Find the number of ordered pairs $(m,n)$ of odd integers greater than $1$ with the property that, in the $m\\times n$ rectangle, the line through the centers of squares $200$ and $2000$ intersects the interior of square $1099$.\n",
      "Found similar problem: Let $m$ and $n$ be odd integers greater than $1.$ An $m\\times n$ rectangle is made up of unit squares where the squares in the top row are numbered left to right with the integers $1$ through $n$, those in the second row are numbered left to right with the integers $n + 1$ through $2n$, and so on. Square $200$ is in the top row, and square $2000$ is in the bottom row. Find the number of ordered pairs $(m,n)$ of odd integers greater than $1$ with the property that, in the $m\\times n$ rectangle, the line through the centers of squares $200$ and $2000$ intersects the interior of square $1099$.\n",
      "\n",
      "1.0\n",
      "Abe can paint the room in 15 hours, Bea can paint 50 percent faster than Abe, and Coe can paint twice as fast as Abe. Abe begins to paint the room and works alone for the first hour and a half. Then Bea joins Abe, and they work together until half the room is painted. Then Coe joins Abe and Bea, and they work together until the entire room is painted. Find the number of minutes after Abe begins for the three of them to finish painting the room.\n",
      "Found similar problem: Abe can paint the room in 15 hours, Bea can paint 50 percent faster than Abe, and Coe can paint twice as fast as Abe. Abe begins to paint the room and works alone for the first hour and a half. Then Bea joins Abe, and they work together until half the room is painted. Then Coe joins Abe and Bea, and they work together until the entire room is painted. Find the number of minutes after Abe begins for the three of them to finish painting the room.\n",
      "\n",
      "1.0\n",
      "A bug walks all day and sleeps all night. On the first day, it starts at point $O$, faces east, and walks a distance of $5$ units due east. Each night the bug rotates $60^\\circ$ counterclockwise. Each day it walks in this new direction half as far as it walked the previous day. The bug gets arbitrarily close to the point $P$. Then $OP^2=\\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n",
      "Found similar problem: A bug walks all day and sleeps all night. On the first day, it starts at point $O$, faces east, and walks a distance of $5$ units due east. Each night the bug rotates $60^\\circ$ counterclockwise. Each day it walks in this new direction half as far as it walked the previous day. The bug gets arbitrarily close to the point $P$. Then $OP^2=\\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n",
      "\n",
      "0.9901670217514038\n",
      "Lilypads $1,2,3,\\ldots$ lie in a row on a pond. A frog makes a sequence of jumps starting on pad $1$. From any pad $k$ the frog jumps to either pad $k+1$ or pad $k+2$ chosen randomly with probability $\\tfrac{1}{2}$ and independently of other jumps. The probability that the frog visits pad $7$ is $\\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.\n",
      "Found similar problem: Lily pads $1,2,3,\\ldots$ lie in a row on a pond. A frog makes a sequence of jumps starting on pad $1$. From any pad $k$ the frog jumps to either pad $k+1$ or pad $k+2$ chosen randomly with probability $\\tfrac{1}{2}$ and independently of other jumps. The probability that the frog visits pad $7$ is $\\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.\n",
      "\n",
      "1.0000001192092896\n",
      "Let $m$ be the number of five-element subsets that can be chosen from the set of the first $14$ natural numbers so that at least two of the five numbers are consecutive. Find the remainder when $m$ is divided by $1000$.\n",
      "Found similar problem: Let $m$ be the number of five-element subsets that can be chosen from the set of the first $14$ natural numbers so that at least two of the five numbers are consecutive. Find the remainder when $m$ is divided by $1000$.\n",
      "\n",
      "0.9258670806884766\n",
      "Given a [nonnegative](https://artofproblemsolving.com/wiki/index.php/Nonnegative) real number $x$, let $\\langle x\\rangle$ denote the fractional part of $x$; that is, $\\langle x\\rangle=x-\\lfloor x\\rfloor$, where $\\lfloor x\\rfloor$ denotes the [greatest integer](https://artofproblemsolving.com/wiki/index.php/Greatest_integer) less than or equal to $x$. Suppose that $a$ is positive, $\\langle a^{-1}\\rangle=\\langle a^2\\rangle$, and $2<a^2<3$. Find the value of $a^{12}-144a^{-1}$.\n",
      "Found similar problem: Given a nonnegative real number $x$, let $\\langle x\\rangle$ denote the fractional part of $x$; that is, $\\langle x\\rangle=x-\\lfloor x\\rfloor$, where $\\lfloor x\\rfloor$ denotes the greatest integer less than or equal to $x$. Suppose that $a$ is positive, $\\langle a^{-1}\\rangle=\\langle a^2\\rangle$, and $2<a^2<3$. Find the value of $a^{12}-144a^{-1}$.\n",
      "\n",
      "1.0\n",
      "There is a unique positive real number $x$ such that the three numbers $\\log_8{2x}$, $\\log_4{x}$, and $\\log_2{x}$, in that order, form a geometric progression with positive common ratio.  The number $x$ can be written as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.\n",
      "Found similar problem: There is a unique positive real number $x$ such that the three numbers $\\log_8{2x}$, $\\log_4{x}$, and $\\log_2{x}$, in that order, form a geometric progression with positive common ratio. The number $x$ can be written as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.\n",
      "\n",
      "1.0\n",
      "For any integer $k\\geq 1$, let $p(k)$ be the smallest prime which does not divide $k.$ Define the integer function $X(k)$ to be the product of all primes less than $p(k)$ if $p(k)>2$, and $X(k)=1$ if $p(k)=2.$ Let $\\{x_n\\}$ be the sequence defined by $x_0=1$, and $x_{n+1}X(x_n)=x_np(x_n)$ for $n\\geq 0.$ Find the smallest positive integer $t$ such that $x_t=2090.$\n",
      "Found similar problem: For any integer $k\\geq 1$, let $p(k)$ be the smallest prime which does not divide $k.$ Define the integer function $X(k)$ to be the product of all primes less than $p(k)$ if $p(k)>2$, and $X(k)=1$ if $p(k)=2.$ Let $\\{x_n\\}$ be the sequence defined by $x_0=1$, and $x_{n+1}X(x_n)=x_np(x_n)$ for $n\\geq 0.$ Find the smallest positive integer $t$ such that $x_t=2090.$\n",
      "\n",
      "1.0\n",
      "Find the number of positive integers less than or equal to $2017$ whose base-three representation contains no digit equal to $0$.\n",
      "Found similar problem: Find the number of positive integers less than or equal to $2017$ whose base-three representation contains no digit equal to $0$.\n",
      "\n",
      "1.0\n",
      "Let $ABCDEF$ be an equiangular hexagon such that $AB=6, BC=8, CD=10$, and $DE=12$. Denote by $d$ the diameter of the largest circle that fits inside the hexagon. Find $d^2$.\n",
      "Found similar problem: Let $ABCDEF$ be an equiangular hexagon such that $AB=6, BC=8, CD=10$, and $DE=12$. Denote by $d$ the diameter of the largest circle that fits inside the hexagon. Find $d^2$.\n",
      "\n",
      "1.0000001192092896\n",
      "At each of the sixteen circles in the network below stands a student. A total of $3360$ coins are distributed among the sixteen students. All at once, all students give away all their coins by passing an equal number of coins to each of their neighbors in the network. After the trade, all students have the same number of coins as they started with. Find the number of coins the student standing at the center circle had originally.\n",
      "\n",
      "[asy] import cse5; unitsize(6mm); defaultpen(linewidth(.8pt)); dotfactor = 8; pathpen=black;  pair A = (0,0); pair B = 2*dir(54), C = 2*dir(126), D = 2*dir(198), E = 2*dir(270), F = 2*dir(342); pair G = 3.6*dir(18), H = 3.6*dir(90), I = 3.6*dir(162), J = 3.6*dir(234), K = 3.6*dir(306); pair M = 6.4*dir(54), N = 6.4*dir(126), O = 6.4*dir(198), P = 6.4*dir(270), L = 6.4*dir(342); pair[] dotted = {A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P};  D(A--B--H--M); D(A--C--H--N); D(A--F--G--L); D(A--E--K--P); D(A--D--J--O); D(B--G--M); D(F--K--L); D(E--J--P); D(O--I--D); D(C--I--N); D(L--M--N--O--P--L);  dot(dotted);  [/asy]\n",
      "Found similar problem: At each of the sixteen circles in the network below stands a student. A total of $3360$ coins are distributed among the sixteen students. All at once, all students give away all their coins by passing an equal number of coins to each of their neighbors in the network. After the trade, all students have the same number of coins as they started with. Find the number of coins the student standing at the center circle had originally.\n",
      "[asy] import cse5; unitsize(6mm); defaultpen(linewidth(.8pt)); dotfactor = 8; pathpen=black; pair A = (0,0); pair B = 2*dir(54), C = 2*dir(126), D = 2*dir(198), E = 2*dir(270), F = 2*dir(342); pair G = 3.6*dir(18), H = 3.6*dir(90), I = 3.6*dir(162), J = 3.6*dir(234), K = 3.6*dir(306); pair M = 6.4*dir(54), N = 6.4*dir(126), O = 6.4*dir(198), P = 6.4*dir(270), L = 6.4*dir(342); pair[] dotted = {A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P}; D(A--B--H--M); D(A--C--H--N); D(A--F--G--L); D(A--E--K--P); D(A--D--J--O); D(B--G--M); D(F--K--L); D(E--J--P); D(O--I--D); D(C--I--N); D(L--M--N--O--P--L); dot(dotted); [/asy]\n",
      "\n",
      "1.0\n",
      "A special deck of cards contains $49$ cards, each labeled with a number from $1$ to $7$ and colored with one of seven colors. Each number-color combination appears on exactly one card. Sharon will select a set of eight cards from the deck at random. Given that she gets at least one card of each color and at least one card with each number, the probability that Sharon can discard one of her cards and $\\textit{still}$ have at least one card of each color and at least one card with each number is $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.\n",
      "Found similar problem: A special deck of cards contains $49$ cards, each labeled with a number from $1$ to $7$ and colored with one of seven colors. Each number-color combination appears on exactly one card. Sharon will select a set of eight cards from the deck at random. Given that she gets at least one card of each color and at least one card with each number, the probability that Sharon can discard one of her cards and $\\textit{still}$ have at least one card of each color and at least one card with each number is $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.\n",
      "\n",
      "0.9995170831680298\n",
      "In a Martian civilization, all logarithms whose bases are not specified are assumed to be base $b$, for some fixed $b\\ge2$. A Martian student writes down\n",
      "\\[3\\log(\\sqrt{x}\\log x)=56\\]\n",
      "\\[\\log_{\\log x}(x)=54\\]\n",
      "and finds that this system of equations has a single real number solution $x>1$. Find $b$.\n",
      "Found similar problem: In a Martian civilization, all logarithms whose bases are not specified as assumed to be base $b$, for some fixed $b\\ge2$. A Martian student writes down \\[3\\log(\\sqrt{x}\\log x)=56\\] \\[\\log_{\\log x}(x)=54\\] and finds that this system of equations has a single real number solution $x>1$. Find $b$.\n",
      "\n",
      "0.9842270016670227\n",
      "There are $N$ [permutations](https://artofproblemsolving.com/wiki/index.php/Permutation) $(a_{1}, a_{2}, ... , a_{30})$ of $1, 2, \\ldots, 30$ such that for $m \\in \\left\\{{2, 3, 5}\\right\\}$, $m$ divides $a_{n+m} - a_{n}$ for all integers $n$ with $1 \\leq n < n+m \\leq 30$. Find the remainder when $N$ is divided by $1000$.\n",
      "Found similar problem: There are $N$ permutations $(a_{1}, a_{2}, ... , a_{30})$ of $1, 2, \\ldots, 30$ such that for $m \\in \\left\\{{2, 3, 5}\\right\\}$, $m$ divides $a_{n+m} - a_{n}$ for all integers $n$ with $1 \\leq n < n+m \\leq 30$. Find the remainder when $N$ is divided by $1000$.\n",
      "\n",
      "1.0\n",
      "For positive real numbers $s$, let $\\tau(s)$ denote the set of all obtuse triangles that have area $s$ and two sides with lengths $4$ and $10$. The set of all $s$ for which $\\tau(s)$ is nonempty, but all triangles in $\\tau(s)$ are congruent, is an interval $[a,b)$. Find $a^2+b^2$.\n",
      "Found similar problem: For positive real numbers $s$, let $\\tau(s)$ denote the set of all obtuse triangles that have area $s$ and two sides with lengths $4$ and $10$. The set of all $s$ for which $\\tau(s)$ is nonempty, but all triangles in $\\tau(s)$ are congruent, is an interval $[a,b)$. Find $a^2+b^2$.\n",
      "\n",
      "1.0\n",
      "Four ambassadors and one advisor for each of them are to be seated at a round table with $12$ chairs numbered in order $1$ to $12$. Each ambassador must sit in an even-numbered chair. Each advisor must sit in a chair adjacent to his or her ambassador. There are $N$ ways for the $8$ people to be seated at the table under these conditions. Find the remainder when $N$ is divided by $1000$.\n",
      "Found similar problem: Four ambassadors and one advisor for each of them are to be seated at a round table with $12$ chairs numbered in order $1$ to $12$. Each ambassador must sit in an even-numbered chair. Each advisor must sit in a chair adjacent to his or her ambassador. There are $N$ ways for the $8$ people to be seated at the table under these conditions. Find the remainder when $N$ is divided by $1000$.\n",
      "\n",
      "1.0\n",
      "In order to complete a large job, $1000$ workers were hired, just enough to complete the job on schedule. All the workers stayed on the job while the first quarter of the work was done, so the first quarter of the work was completed on schedule. Then $100$ workers were laid off, so the second quarter of the work was completed behind schedule. Then an additional $100$ workers were laid off, so the third quarter of the work was completed still further behind schedule. Given that all workers work at the same rate, what is the minimum number of additional workers, beyond the $800$ workers still on the job at the end of the third quarter, that must be hired after three-quarters of the work has been completed so that the entire project can be completed on schedule or before?\n",
      "Found similar problem: In order to complete a large job, $1000$ workers were hired, just enough to complete the job on schedule. All the workers stayed on the job while the first quarter of the work was done, so the first quarter of the work was completed on schedule. Then $100$ workers were laid off, so the second quarter of the work was completed behind schedule. Then an additional $100$ workers were laid off, so the third quarter of the work was completed still further behind schedule. Given that all workers work at the same rate, what is the minimum number of additional workers, beyond the $800$ workers still on the job at the end of the third quarter, that must be hired after three-quarters of the work has been completed so that the entire project can be completed on schedule or before?\n",
      "\n",
      "1.0\n",
      "Given that $x, y,$ and $z$ are real numbers that satisfy:\n",
      "\\begin{align*} x &= \\sqrt{y^2-\\frac{1}{16}}+\\sqrt{z^2-\\frac{1}{16}}, \\\\ y &= \\sqrt{z^2-\\frac{1}{25}}+\\sqrt{x^2-\\frac{1}{25}}, \\\\ z &= \\sqrt{x^2 - \\frac 1{36}}+\\sqrt{y^2-\\frac 1{36}}, \\end{align*}\n",
      "and that $x+y+z = \\frac{m}{\\sqrt{n}},$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime, find $m+n.$\n",
      "Found similar problem: Given that $x, y,$ and $z$ are real numbers that satisfy: \\begin{align*} x &= \\sqrt{y^2-\\frac{1}{16}}+\\sqrt{z^2-\\frac{1}{16}}, \\\\ y &= \\sqrt{z^2-\\frac{1}{25}}+\\sqrt{x^2-\\frac{1}{25}}, \\\\ z &= \\sqrt{x^2 - \\frac 1{36}}+\\sqrt{y^2-\\frac 1{36}}, \\end{align*} and that $x+y+z = \\frac{m}{\\sqrt{n}},$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime, find $m+n.$\n",
      "\n",
      "1.0\n",
      "For integers $a,b,c$ and $d,$ let $f(x)=x^2+ax+b$ and $g(x)=x^2+cx+d.$ Find the number of ordered triples $(a,b,c)$ of integers with absolute values not exceeding $10$ for which there is an integer $d$ such that $g(f(2))=g(f(4))=0.$\n",
      "Found similar problem: For integers $a,b,c$ and $d,$ let $f(x)=x^2+ax+b$ and $g(x)=x^2+cx+d.$ Find the number of ordered triples $(a,b,c)$ of integers with absolute values not exceeding $10$ for which there is an integer $d$ such that $g(f(2))=g(f(4))=0.$\n",
      "\n",
      "1.0\n",
      "The AIME Triathlon consists of a half-mile swim, a 30-mile bicycle ride, and an eight-mile run. Tom swims, bicycles, and runs at constant rates. He runs fives times as fast as he swims, and he bicycles twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend bicycling?\n",
      "Found similar problem: The AIME Triathlon consists of a half-mile swim, a 30-mile bicycle ride, and an eight-mile run. Tom swims, bicycles, and runs at constant rates. He runs fives times as fast as he swims, and he bicycles twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend bicycling?\n",
      "\n",
      "0.9969760179519653\n",
      "For distinct complex numbers $z_1,z_2,\\dots,z_{673}$, the polynomial\n",
      "\\[(x-z_1)^3(x-z_2)^3 \\cdots (x-z_{673})^3\\]can be expressed as $x^{2019} + 20x^{2018} + 19x^{2017}+g(x)$, where $g(x)$ is a polynomial with complex coefficients and with degree at most $2016$. The sum \\[\\left| \\sum_{1 \\le j <k \\le 673} z_jz_k \\right|\\] can be expressed in the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n",
      "Found similar problem: For distinct complex numbers $z_1,z_2,\\dots,z_{673}$, the polynomial \\[(x-z_1)^3(x-z_2)^3 \\cdots (x-z_{673})^3\\]can be expressed as $x^{2019} + 20x^{2018} + 19x^{2017}+g(x)$, where $g(x)$ is a polynomial with complex coefficients and with degree at most $2016$. The sum $\\left| \\sum_{1 \\le j <k \\le 673} z_jz_k \\right|$ can be expressed in the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n",
      "\n",
      "0.9985167384147644\n",
      "Find the number of sets $\\{a,b,c\\}$ of three distinct positive integers with the property that the product of $a,b,$ and $c$ is equal to the product of $11,21,31,41,51,61$.\n",
      "Found similar problem: Find the number of sets ${a,b,c}$ of three distinct positive integers with the property that the product of $a,b,$ and $c$ is equal to the product of $11,21,31,41,51,61$.\n",
      "\n",
      "1.0\n",
      "Jar A contains four liters of a solution that is 45% acid.  Jar B contains five liters of a solution that is 48% acid.  Jar C contains one liter of a solution that is $k\\%$ acid.  From jar C, $\\frac{m}{n}$ liters of the solution is added to jar A, and the remainder of the solution in jar C is added to jar B.  At the end both jar A and jar B contain solutions that are 50% acid.  Given that $m$ and $n$ are relatively prime positive integers, find $k + m + n$.\n",
      "Found similar problem: Jar A contains four liters of a solution that is 45% acid. Jar B contains five liters of a solution that is 48% acid. Jar C contains one liter of a solution that is $k\\%$ acid. From jar C, $\\frac{m}{n}$ liters of the solution is added to jar A, and the remainder of the solution in jar C is added to jar B. At the end both jar A and jar B contain solutions that are 50% acid. Given that $m$ and $n$ are relatively prime positive integers, find $k + m + n$.\n",
      "\n",
      "0.9184293150901794\n",
      "For how many [ordered pairs](https://artofproblemsolving.com/wiki/index.php/Ordered_pair) $(x,y)$ of [integers](https://artofproblemsolving.com/wiki/index.php/Integer) is it true that $0 < x < y < 10^6$ and that the [arithmetic mean](https://artofproblemsolving.com/wiki/index.php/Arithmetic_mean) of $x$ and $y$ is exactly $2$ more than the [geometric mean](https://artofproblemsolving.com/wiki/index.php/Geometric_mean) of $x$ and $y$?\n",
      "Found similar problem: For how many ordered pairs $(x,y)$ of integers is it true that $0 < x < y < 10^6$ and that the arithmetic mean of $x$ and $y$ is exactly $2$ more than the geometric mean of $x$ and $y$?\n",
      "\n",
      "0.9650360941886902\n",
      "Let $K$ be the product of all factors $(b-a)$ (not necessarily distinct) where $a$ and $b$ are integers satisfying $1\\le a < b \\le 20$. Find the greatest positive [integer](https://artofproblemsolving.com/wiki/index.php/Integer) $n$ such that $2^n$ divides $K$.\n",
      "Found similar problem: Let $K$ be the product of all factors $(b-a)$ (not necessarily distinct) where $a$ and $b$ are integers satisfying $1\\le a < b \\le 20$. Find the greatest positive integer $n$ such that $2^n$ divides $K$.\n",
      "\n",
      "0.9215917587280273\n",
      "The shortest distances between an interior [diagonal](https://artofproblemsolving.com/wiki/index.php/Diagonal) of a rectangular [parallelepiped](https://artofproblemsolving.com/wiki/index.php/Parallelepiped), $P$, and the edges it does not meet are $2\\sqrt{5}$, $\\frac{30}{\\sqrt{13}}$, and $\\frac{15}{\\sqrt{10}}$. Determine the [volume](https://artofproblemsolving.com/wiki/index.php/Volume) of $P$.\n",
      "Found similar problem: The shortest distances between an interior diagonal of a rectangular parallelepiped, $P$, and the edges it does not meet are $2\\sqrt{5}$, $\\frac{30}{\\sqrt{13}}$, and $\\frac{15}{\\sqrt{10}}$. Determine the volume of $P$.\n",
      "\n",
      "1.0\n",
      "Dave rolls a fair six-sided die until a six appears for the first time. Independently, Linda rolls a fair six-sided die until a six appears for the first time. Let $m$ and $n$ be relatively prime positive integers such that $\\dfrac mn$ is the probability that the number of times Dave rolls his die is equal to or within one of the number of times Linda rolls her die. Find $m+n$.\n",
      "Found similar problem: Dave rolls a fair six-sided die until a six appears for the first time. Independently, Linda rolls a fair six-sided die until a six appears for the first time. Let $m$ and $n$ be relatively prime positive integers such that $\\dfrac mn$ is the probability that the number of times Dave rolls his die is equal to or within one of the number of times Linda rolls her die. Find $m+n$.\n",
      "\n",
      "1.0\n",
      "For $n \\ge 1$ call a finite sequence $(a_1, a_2 \\ldots a_n)$ of positive integers progressive if $a_i < a_{i+1}$ and $a_i$ divides $a_{i+1}$ for all $1 \\le i \\le n-1$. Find the number of progressive sequences such that the sum of the terms in the sequence is equal to $360$.\n",
      "Found similar problem: For $n \\ge 1$ call a finite sequence $(a_1, a_2 \\ldots a_n)$ of positive integers progressive if $a_i < a_{i+1}$ and $a_i$ divides $a_{i+1}$ for all $1 \\le i \\le n-1$. Find the number of progressive sequences such that the sum of the terms in the sequence is equal to $360$.\n",
      "\n",
      "1.0000001192092896\n",
      "Let $ABCD$ be a square, and let $E$ and $F$ be points on $\\overline{AB}$ and $\\overline{BC},$ respectively. The line through $E$ parallel to $\\overline{BC}$ and the line through $F$ parallel to $\\overline{AB}$ divide $ABCD$ into two squares and two nonsquare rectangles. The sum of the areas of the two squares is $\\frac{9}{10}$ of the area of square $ABCD.$ Find $\\frac{AE}{EB} + \\frac{EB}{AE}.$\n",
      "Found similar problem: Let $ABCD$ be a square, and let $E$ and $F$ be points on $\\overline{AB}$ and $\\overline{BC},$ respectively. The line through $E$ parallel to $\\overline{BC}$ and the line through $F$ parallel to $\\overline{AB}$ divide $ABCD$ into two squares and two nonsquare rectangles. The sum of the areas of the two squares is $\\frac{9}{10}$ of the area of square $ABCD.$ Find $\\frac{AE}{EB} + \\frac{EB}{AE}.$\n",
      "\n",
      "1.0\n",
      "Positive numbers $x$, $y$, and $z$ satisfy $xyz = 10^{81}$ and $(\\log_{10}x)(\\log_{10} yz) + (\\log_{10}y) (\\log_{10}z) = 468$. Find $\\sqrt {(\\log_{10}x)^2 + (\\log_{10}y)^2 + (\\log_{10}z)^2}$.\n",
      "Found similar problem: Positive numbers $x$, $y$, and $z$ satisfy $xyz = 10^{81}$ and $(\\log_{10}x)(\\log_{10} yz) + (\\log_{10}y) (\\log_{10}z) = 468$. Find $\\sqrt {(\\log_{10}x)^2 + (\\log_{10}y)^2 + (\\log_{10}z)^2}$.\n",
      "\n",
      "1.000000238418579\n",
      "Define a T-grid to be a $3\\times3$ matrix which satisfies the following two properties:\n",
      "\n",
      "\n",
      "Exactly five of the entries are $1$'s, and the remaining four entries are $0$'s.\n",
      "Among the eight rows, columns, and long diagonals (the long diagonals are $\\{a_{13},a_{22},a_{31}\\}$ and $\\{a_{11},a_{22},a_{33}\\}$, no more than one of the eight has all three entries equal.\n",
      "Find the number of distinct T-grids.\n",
      "Found similar problem: Define a T-grid to be a $3\\times3$ matrix which satisfies the following two properties:\n",
      "Exactly five of the entries are $1$'s, and the remaining four entries are $0$'s.\n",
      "Among the eight rows, columns, and long diagonals (the long diagonals are $\\{a_{13},a_{22},a_{31}\\}$ and $\\{a_{11},a_{22},a_{33}\\}$, no more than one of the eight has all three entries equal.\n",
      " Find the number of distinct T-grids.\n",
      "\n",
      "1.0000001192092896\n",
      "Find the least positive integer $N$ such that the set of $1000$ consecutive integers beginning with $1000\\cdot N$ contains no square of an integer.\n",
      "Found similar problem: Find the least positive integer $N$ such that the set of $1000$ consecutive integers beginning with $1000\\cdot N$ contains no square of an integer.\n",
      "\n",
      "1.0\n",
      "Find the least positive integer $n$ for which $2^n + 5^n - n$ is a multiple of $1000$.\n",
      "Found similar problem: Find the least positive integer $n$ for which $2^n + 5^n - n$ is a multiple of $1000$.\n",
      "\n",
      "1.0\n",
      "When each of $702$, $787$, and $855$ is divided by the positive integer $m$, the remainder is always the positive integer $r$. When each of $412$, $722$, and $815$ is divided by the positive integer $n$, the remainder is always the positive integer $s \\neq r$. Find $m+n+r+s$.\n",
      "Found similar problem: When each of $702$, $787$, and $855$ is divided by the positive integer $m$, the remainder is always the positive integer $r$. When each of $412$, $722$, and $815$ is divided by the positive integer $n$, the remainder is always the positive integer $s \\neq r$. Find $m+n+r+s$.\n",
      "\n",
      "1.0000001192092896\n",
      "A flat board has a circular hole with radius $1$ and a circular hole with radius $2$ such that the distance between the centers of the two holes is $7.$ Two spheres with equal radii sit in the two holes such that the spheres are tangent to each other. The square of the radius of the spheres is $\\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$\n",
      "Found similar problem: A flat board has a circular hole with radius $1$ and a circular hole with radius $2$ such that the distance between the centers of the two holes is $7.$ Two spheres with equal radii sit in the two holes such that the spheres are tangent to each other. The square of the radius of the spheres is $\\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$\n",
      "\n",
      "1.0\n",
      "Let $N$ be the largest positive integer with the following property: reading from left to right, each pair of consecutive digits of $N$ forms a perfect square. What are the leftmost three digits of $N$?\n",
      "Found similar problem: Let $N$ be the largest positive integer with the following property: reading from left to right, each pair of consecutive digits of $N$ forms a perfect square. What are the leftmost three digits of $N$?\n",
      "\n",
      "1.0\n",
      "Let $x,$ $y,$ and $z$ be positive real numbers that satisfy\n",
      "\\[2\\log_{x}(2y) = 2\\log_{2x}(4z) = \\log_{2x^4}(8yz) \\ne 0.\\]\n",
      "The value of $xy^5z$ can be expressed in the form $\\frac{1}{2^{p/q}},$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$\n",
      "Found similar problem: Let $x,$ $y,$ and $z$ be positive real numbers that satisfy \\[2\\log_{x}(2y) = 2\\log_{2x}(4z) = \\log_{2x^4}(8yz) \\ne 0.\\] The value of $xy^5z$ can be expressed in the form $\\frac{1}{2^{p/q}},$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$\n",
      "\n",
      "1.0000001192092896\n",
      "Alfred and Bonnie play a game in which they take turns tossing a fair coin. The winner of a game is the first person to obtain a head. Alfred and Bonnie play this game several times with the stipulation that the loser of a game goes first in the next game. Suppose that Alfred goes first in the first game, and that the probability that he wins the sixth game is $m/n\\,$, where $m\\,$ and $n\\,$ are relatively prime positive integers. What are the last three digits of $m+n\\,$?\n",
      "Found similar problem: Alfred and Bonnie play a game in which they take turns tossing a fair coin. The winner of a game is the first person to obtain a head. Alfred and Bonnie play this game several times with the stipulation that the loser of a game goes first in the next game. Suppose that Alfred goes first in the first game, and that the probability that he wins the sixth game is $m/n\\,$, where $m\\,$ and $n\\,$ are relatively prime positive integers. What are the last three digits of $m+n\\,$?\n",
      "\n",
      "1.0\n",
      "Find the number of functions $f(x)$ from $\\{1, 2, 3, 4, 5\\}$ to $\\{1, 2, 3, 4, 5\\}$ that satisfy $f(f(x)) = f(f(f(x)))$ for all $x$ in $\\{1, 2, 3, 4, 5\\}$.\n",
      "Found similar problem: Find the number of functions $f(x)$ from $\\{1, 2, 3, 4, 5\\}$ to $\\{1, 2, 3, 4, 5\\}$ that satisfy $f(f(x)) = f(f(f(x)))$ for all $x$ in $\\{1, 2, 3, 4, 5\\}$.\n",
      "\n",
      "0.9658002257347107\n",
      "For all positive integers $x$, let\n",
      "\\[f(x)=\\begin{cases}1 & \\text{if }x = 1\\\\ \\frac x{10} & \\text{if }x\\text{ is divisible by 10}\\\\ x+1 & \\text{otherwise}\\end{cases}\\]\n",
      "and define a [sequence](https://artofproblemsolving.com/wiki/index.php/Sequence) as follows: $x_1=x$ and $x_{n+1}=f(x_n)$ for all positive integers $n$. Let $d(x)$ be the smallest $n$ such that $x_n=1$. (For example, $d(100)=3$ and $d(87)=7$.) Let $m$ be the number of positive integers $x$ such that $d(x)=20$. Find the sum of the distinct prime factors of $m$.\n",
      "Found similar problem: For all positive integers $x$, let \\[f(x)=\\begin{cases}1 & \\text{if }x = 1\\\\ \\frac x{10} & \\text{if }x\\text{ is divisible by 10}\\\\ x+1 & \\text{otherwise}\\end{cases}\\] and define a sequence as follows: $x_1=x$ and $x_{n+1}=f(x_n)$ for all positive integers $n$. Let $d(x)$ be the smallest $n$ such that $x_n=1$. (For example, $d(100)=3$ and $d(87)=7$.) Let $m$ be the number of positive integers $x$ such that $d(x)=20$. Find the sum of the distinct prime factors of $m$.\n",
      "\n",
      "1.0\n",
      "Let the sum of a set of numbers be the sum of its elements. Let $S$ be a set of positive integers, none greater than 15. Suppose no two disjoint subsets of $S$ have the same sum. What is the largest sum a set $S$ with these properties can have?\n",
      "Found similar problem: Let the sum of a set of numbers be the sum of its elements. Let $S$ be a set of positive integers, none greater than 15. Suppose no two disjoint subsets of $S$ have the same sum. What is the largest sum a set $S$ with these properties can have?\n",
      "\n",
      "0.9495150446891785\n",
      "Let $m/n$, in lowest terms, be the [probability](https://artofproblemsolving.com/wiki/index.php/Probability) that a randomly chosen positive [divisor](https://artofproblemsolving.com/wiki/index.php/Divisor) of $10^{99}$ is an integer multiple of $10^{88}$.  Find $m + n$.\n",
      "Found similar problem: Let $m/n$, in lowest terms, be the probability that a randomly chosen positive divisor of $10^{99}$ is an integer multiple of $10^{88}$. Find $m + n$.\n",
      "\n",
      "1.0000001192092896\n",
      "A frog begins at $P_0 = (0,0)$ and makes a sequence of jumps according to the following rule: from $P_n = (x_n, y_n),$ the frog jumps to $P_{n+1},$ which may be any of the points $(x_n + 7, y_n + 2),$ $(x_n + 2, y_n + 7),$ $(x_n - 5, y_n - 10),$ or $(x_n - 10, y_n - 5).$ There are $M$ points $(x, y)$ with $|x| + |y| \\le 100$ that can be reached by a sequence of such jumps. Find the remainder when $M$ is divided by $1000.$\n",
      "Found similar problem: A frog begins at $P_0 = (0,0)$ and makes a sequence of jumps according to the following rule: from $P_n = (x_n, y_n),$ the frog jumps to $P_{n+1},$ which may be any of the points $(x_n + 7, y_n + 2),$ $(x_n + 2, y_n + 7),$ $(x_n - 5, y_n - 10),$ or $(x_n - 10, y_n - 5).$ There are $M$ points $(x, y)$ with $|x| + |y| \\le 100$ that can be reached by a sequence of such jumps. Find the remainder when $M$ is divided by $1000.$\n",
      "\n",
      "1.0\n",
      "There is a unique angle $\\theta$ between $0^{\\circ}$ and $90^{\\circ}$ such that for nonnegative integers $n$, the value of $\\tan{\\left(2^{n}\\theta\\right)}$ is positive when $n$ is a multiple of $3$, and negative otherwise. The degree measure of $\\theta$ is $\\tfrac{p}{q}$, where $p$ and $q$ are relatively prime integers. Find $p+q$.\n",
      "Found similar problem: There is a unique angle $\\theta$ between $0^{\\circ}$ and $90^{\\circ}$ such that for nonnegative integers $n$, the value of $\\tan{\\left(2^{n}\\theta\\right)}$ is positive when $n$ is a multiple of $3$, and negative otherwise. The degree measure of $\\theta$ is $\\tfrac{p}{q}$, where $p$ and $q$ are relatively prime integers. Find $p+q$.\n",
      "\n",
      "1.0\n",
      "An $a \\times b \\times c$ rectangular box is built from $a \\cdot b \\cdot c$ unit cubes. Each unit cube is colored red, green, or yellow. Each of the $a$ layers of size $1 \\times b \\times c$ parallel to the $(b \\times c)$ faces of the box contains exactly $9$ red cubes, exactly $12$ green cubes, and some yellow cubes. Each of the $b$ layers of size $a \\times 1 \\times c$ parallel to the $(a \\times c)$ faces of the box contains exactly $20$ green cubes, exactly $25$ yellow cubes, and some red cubes. Find the smallest possible volume of the box.\n",
      "Found similar problem: An $a \\times b \\times c$ rectangular box is built from $a \\cdot b \\cdot c$ unit cubes. Each unit cube is colored red, green, or yellow. Each of the $a$ layers of size $1 \\times b \\times c$ parallel to the $(b \\times c)$ faces of the box contains exactly $9$ red cubes, exactly $12$ green cubes, and some yellow cubes. Each of the $b$ layers of size $a \\times 1 \\times c$ parallel to the $(a \\times c)$ faces of the box contains exactly $20$ green cubes, exactly $25$ yellow cubes, and some red cubes. Find the smallest possible volume of the box.\n",
      "\n",
      "1.0\n",
      "Find the smallest integer $k$ for which the conditions\n",
      "(1) $a_1,a_2,a_3\\cdots$ is a nondecreasing sequence of positive integers\n",
      "(2) $a_n=a_{n-1}+a_{n-2}$ for all $n>2$\n",
      "(3) $a_9=k$\n",
      "are satisfied by more than one sequence.\n",
      "Found similar problem: Find the smallest integer $k$ for which the conditions\n",
      " (1) $a_1,a_2,a_3\\cdots$ is a nondecreasing sequence of positive integers\n",
      " (2) $a_n=a_{n-1}+a_{n-2}$ for all $n>2$\n",
      " (3) $a_9=k$\n",
      " are satisfied by more than one sequence.\n",
      "\n",
      "1.0\n",
      "A rectangle that is inscribed in a larger rectangle (with one vertex on each side) is called unstuck if it is possible to rotate (however slightly) the smaller rectangle about its center within the confines of the larger. Of all the rectangles that can be inscribed unstuck in a 6 by 8 rectangle, the smallest perimeter has the form $\\sqrt{N}\\,$, for a positive integer $N\\,$. Find $N\\,$.\n",
      "Found similar problem: A rectangle that is inscribed in a larger rectangle (with one vertex on each side) is called unstuck if it is possible to rotate (however slightly) the smaller rectangle about its center within the confines of the larger. Of all the rectangles that can be inscribed unstuck in a 6 by 8 rectangle, the smallest perimeter has the form $\\sqrt{N}\\,$, for a positive integer $N\\,$. Find $N\\,$.\n",
      "\n",
      "0.9778977632522583\n",
      "Gary purchased a large beverage, but only drank $m/n$ of it, where $m$ and $n$ are [relatively prime](https://artofproblemsolving.com/wiki/index.php/Relatively_prime) positive integers. If he had purchased half as much and drunk twice as much, he would have wasted only $2/9$ as much beverage. Find $m+n$.\n",
      "Found similar problem: Gary purchased a large beverage, but only drank $m/n$ of it, where $m$ and $n$ are relatively prime positive integers. If he had purchased half as much and drunk twice as much, he would have wasted only $2/9$ as much beverage. Find $m+n$.\n",
      "\n",
      "0.960173487663269\n",
      "A [sequence](https://artofproblemsolving.com/wiki/index.php/Sequence) of numbers $x_{1},x_{2},x_{3},\\ldots,x_{100}$ has the property that, for every [integer](https://artofproblemsolving.com/wiki/index.php/Integer) $k$ between $1$ and $100,$ inclusive, the number $x_{k}$ is $k$ less than the sum of the other $99$ numbers. Given that $x_{50} = m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m + n$.\n",
      "Found similar problem: A sequence of numbers $x_{1},x_{2},x_{3},\\ldots,x_{100}$ has the property that, for every integer $k$ between $1$ and $100,$ inclusive, the number $x_{k}$ is $k$ less than the sum of the other $99$ numbers. Given that $x_{50} = m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m + n$.\n",
      "\n",
      "0.960322916507721\n",
      "Find the smallest positive [integer](https://artofproblemsolving.com/wiki/index.php/Integer) $n$ for which the expansion of $(xy-3x+7y-21)^n$, after like terms have been collected, has at least 1996 terms.\n",
      "Found similar problem: Find the smallest positive integer $n$ for which the expansion of $(xy-3x+7y-21)^n$, after like terms have been collected, has at least 1996 terms.\n",
      "\n",
      "1.0000001192092896\n",
      "Given $f(z) = z^2-19z$, there are complex numbers $z$ with the property that $z$, $f(z)$, and $f(f(z))$ are the vertices of a right triangle in the complex plane with a right angle at $f(z)$. There are positive integers $m$ and $n$ such that one such value of $z$ is $m+\\sqrt{n}+11i$. Find $m+n$.\n",
      "Found similar problem: Given $f(z) = z^2-19z$, there are complex numbers $z$ with the property that $z$, $f(z)$, and $f(f(z))$ are the vertices of a right triangle in the complex plane with a right angle at $f(z)$. There are positive integers $m$ and $n$ such that one such value of $z$ is $m+\\sqrt{n}+11i$. Find $m+n$.\n",
      "\n",
      "1.0\n",
      "The teams $T_1$, $T_2$, $T_3$, and $T_4$ are in the playoffs. In the semifinal matches, $T_1$ plays $T_4$, and $T_2$ plays $T_3$. The winners of those two matches will play each other in the final match to determine the champion. When $T_i$ plays $T_j$, the probability that $T_i$ wins is $\\frac{i}{i+j}$, and the outcomes of all the matches are independent. The probability that $T_4$ will be the champion is $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.\n",
      "Found similar problem: The teams $T_1$, $T_2$, $T_3$, and $T_4$ are in the playoffs. In the semifinal matches, $T_1$ plays $T_4$, and $T_2$ plays $T_3$. The winners of those two matches will play each other in the final match to determine the champion. When $T_i$ plays $T_j$, the probability that $T_i$ wins is $\\frac{i}{i+j}$, and the outcomes of all the matches are independent. The probability that $T_4$ will be the champion is $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.\n",
      "\n",
      "1.0\n",
      "Butch and Sundance need to get out of Dodge. To travel as quickly as possible, each alternates walking and riding their only horse, Sparky, as follows. Butch begins by walking while Sundance rides. When Sundance reaches the first of the hitching posts that are conveniently located at one-mile intervals along their route, he ties Sparky to the post and begins walking. When Butch reaches Sparky, he rides until he passes Sundance, then leaves Sparky at the next hitching post and resumes walking, and they continue in this manner. Sparky, Butch, and Sundance walk at $6,$ $4,$ and $2.5$ miles per hour, respectively. The first time Butch and Sundance meet at a milepost, they are $n$ miles from Dodge, and they have been traveling for $t$ minutes. Find $n + t$.\n",
      "Found similar problem: Butch and Sundance need to get out of Dodge. To travel as quickly as possible, each alternates walking and riding their only horse, Sparky, as follows. Butch begins by walking while Sundance rides. When Sundance reaches the first of the hitching posts that are conveniently located at one-mile intervals along their route, he ties Sparky to the post and begins walking. When Butch reaches Sparky, he rides until he passes Sundance, then leaves Sparky at the next hitching post and resumes walking, and they continue in this manner. Sparky, Butch, and Sundance walk at $6,$ $4,$ and $2.5$ miles per hour, respectively. The first time Butch and Sundance meet at a milepost, they are $n$ miles from Dodge, and they have been traveling for $t$ minutes. Find $n + t$.\n",
      "\n",
      "1.0\n",
      "Let $B$ be the set of all binary integers that can be written using exactly $5$ zeros and $8$ ones where leading zeros are allowed. If all possible subtractions are performed in which one element of $B$ is subtracted from another, find the number of times the answer $1$ is obtained.\n",
      "Found similar problem: Let $B$ be the set of all binary integers that can be written using exactly $5$ zeros and $8$ ones where leading zeros are allowed. If all possible subtractions are performed in which one element of $B$ is subtracted from another, find the number of times the answer $1$ is obtained.\n",
      "\n",
      "0.7896944284439087\n",
      "[Rectangle](https://artofproblemsolving.com/wiki/index.php/Rectangle) $ABCD$ is divided into four parts of equal [area](https://artofproblemsolving.com/wiki/index.php/Area) by five [ segments](https://artofproblemsolving.com/wiki/index.php/Line_segment) as shown in the figure, where $XY = YB + BC + CZ = ZW = WD + DA + AX$, and $PQ$ is [parallel](https://artofproblemsolving.com/wiki/index.php/Parallel) to $AB$.  Find the [length](https://artofproblemsolving.com/wiki/index.php/Length) of $AB$ (in cm) if $BC = 19$ cm and $PQ = 87$ cm.\n",
      "[AIME 1987 Problem 6.png](https://artofproblemsolving.com/wiki/index.php/File:AIME_1987_Problem_6.png)\n",
      "Found similar problem: Rectangle $ABCD$ is divided into four parts of equal area by five segments as shown in the figure, where $XY = YB + BC + CZ = ZW = WD + DA + AX$, and $PQ$ is parallel to $AB$. Find the length of $AB$ (in cm) if $BC = 19$ cm and $PQ = 87$ cm.\n",
      "\n",
      "1.0\n",
      "In an isosceles trapezoid, the parallel bases have lengths $\\log 3$ and $\\log 192$, and the altitude to these bases has length $\\log 16$. The perimeter of the trapezoid can be written in the form $\\log 2^p 3^q$, where $p$ and $q$ are positive integers. Find $p + q$.\n",
      "Found similar problem: In an isosceles trapezoid, the parallel bases have lengths $\\log 3$ and $\\log 192$, and the altitude to these bases has length $\\log 16$. The perimeter of the trapezoid can be written in the form $\\log 2^p 3^q$, where $p$ and $q$ are positive integers. Find $p + q$.\n",
      "\n",
      "0.9662343263626099\n",
      "A 100 foot long moving walkway moves at a constant rate of 6 feet per second.  Al steps onto the start of the walkway and stands.  Bob steps onto the start of the walkway two seconds later and strolls forward along the walkway at a constant rate of 4 feet per second.  Two seconds after that, Cy reaches the start of the walkway and walks briskly forward beside the walkway at a constant rate of 8 feet per second.  At a certain time, one of these three persons is exactly halfway between the other two.  At that time, find the [distance](https://artofproblemsolving.com/wiki/index.php/Distance) in feet between the start of the walkway and the middle person.\n",
      "Found similar problem: A 100 foot long moving walkway moves at a constant rate of 6 feet per second. Al steps onto the start of the walkway and stands. Bob steps onto the start of the walkway two seconds later and strolls forward along the walkway at a constant rate of 4 feet per second. Two seconds after that, Cy reaches the start of the walkway and walks briskly forward beside the walkway at a constant rate of 8 feet per second. At a certain time, one of these three persons is exactly halfway between the other two. At that time, find the distance in feet between the start of the walkway and the middle person.\n",
      "\n",
      "1.0\n",
      "A teacher was leading a class of four perfectly logical students. The teacher chose a set $S$ of four integers and gave a different number in $S$ to each student. Then the teacher announced to the class that the numbers in $S$ were four consecutive two-digit positive integers, that some number in $S$ was divisible by $6$, and a different number in $S$ was divisible by $7$. The teacher then asked if any of the students could deduce what $S$ is, but in unison, all of the students replied no.\n",
      "However, upon hearing that all four students replied no, each student was able to determine the elements of $S$. Find the sum of all possible values of the greatest element of $S$.\n",
      "Found similar problem: A teacher was leading a class of four perfectly logical students. The teacher chose a set $S$ of four integers and gave a different number in $S$ to each student. Then the teacher announced to the class that the numbers in $S$ were four consecutive two-digit positive integers, that some number in $S$ was divisible by $6$, and a different number in $S$ was divisible by $7$. The teacher then asked if any of the students could deduce what $S$ is, but in unison, all of the students replied no.\n",
      " However, upon hearing that all four students replied no, each student was able to determine the elements of $S$. Find the sum of all possible values of the greatest element of $S$.\n",
      "\n",
      "1.0000001192092896\n",
      "A $10\\times10\\times10$ grid of points consists of all points in space of the form $(i,j,k)$, where $i$, $j$, and $k$ are integers between $1$ and $10$, inclusive. Find the number of different lines that contain exactly $8$ of these points.\n",
      "Found similar problem: A $10\\times10\\times10$ grid of points consists of all points in space of the form $(i,j,k)$, where $i$, $j$, and $k$ are integers between $1$ and $10$, inclusive. Find the number of different lines that contain exactly $8$ of these points.\n",
      "\n",
      "1.0\n",
      "Let $n$ be the least positive integer for which $149^n-2^n$ is divisible by $3^3\\cdot5^5\\cdot7^7.$ Find the number of positive integer divisors of $n.$\n",
      "Found similar problem: Let $n$ be the least positive integer for which $149^n-2^n$ is divisible by $3^3\\cdot5^5\\cdot7^7.$ Find the number of positive integer divisors of $n.$\n",
      "\n",
      "1.0\n",
      "Triangle $ABC$ has $AB=40,AC=31,$ and $\\sin{A}=\\frac{1}{5}$. This triangle is inscribed in rectangle $AQRS$ with $B$ on $\\overline{QR}$ and $C$ on $\\overline{RS}$. Find the maximum possible area of $AQRS$.\n",
      "Found similar problem: Triangle $ABC$ has $AB=40,AC=31,$ and $\\sin{A}=\\frac{1}{5}$. This triangle is inscribed in rectangle $AQRS$ with $B$ on $\\overline{QR}$ and $C$ on $\\overline{RS}$. Find the maximum possible area of $AQRS$.\n",
      "\n",
      "1.0\n",
      "The vertices of a regular nonagon (9-sided polygon) are to be labeled with the digits 1 through 9 in such a way that the sum of the numbers on every three consecutive vertices is a multiple of 3.  Two acceptable arrangements are considered to be indistinguishable if one can be obtained from the other by rotating the nonagon in the plane.  Find the number of distinguishable acceptable arrangements.\n",
      "Found similar problem: The vertices of a regular nonagon (9-sided polygon) are to be labeled with the digits 1 through 9 in such a way that the sum of the numbers on every three consecutive vertices is a multiple of 3. Two acceptable arrangements are considered to be indistinguishable if one can be obtained from the other by rotating the nonagon in the plane. Find the number of distinguishable acceptable arrangements.\n",
      "\n",
      "1.0\n",
      "In the accompanying figure, the outer square $S$ has side length $40$. A second square $S'$ of side length $15$ is constructed inside $S$ with the same center as $S$ and with sides parallel to those of $S$. From each midpoint of a side of $S$, segments are drawn to the two closest vertices of $S'$. The result is a four-pointed starlike figure inscribed in $S$. The star figure is cut out and then folded to form a pyramid with base $S'$. Find the volume of this pyramid.\n",
      "\n",
      "[asy]  pair S1 = (20, 20), S2 = (-20, 20), S3 = (-20, -20), S4 = (20, -20); pair M1 = (S1+S2)/2, M2 = (S2+S3)/2, M3=(S3+S4)/2, M4=(S4+S1)/2; pair Sp1 = (7.5, 7.5), Sp2=(-7.5, 7.5), Sp3 = (-7.5, -7.5), Sp4 = (7.5, -7.5);  draw(S1--S2--S3--S4--cycle); draw(Sp1--Sp2--Sp3--Sp4--cycle); draw(Sp1--M1--Sp2--M2--Sp3--M3--Sp4--M4--cycle); [/asy]\n",
      "Found similar problem: In the accompanying figure, the outer square $S$ has side length $40$. A second square $S'$ of side length $15$ is constructed inside $S$ with the same center as $S$ and with sides parallel to those of $S$. From each midpoint of a side of $S$, segments are drawn to the two closest vertices of $S'$. The result is a four-pointed starlike figure inscribed in $S$. The star figure is cut out and then folded to form a pyramid with base $S'$. Find the volume of this pyramid.\n",
      "[asy] pair S1 = (20, 20), S2 = (-20, 20), S3 = (-20, -20), S4 = (20, -20); pair M1 = (S1+S2)/2, M2 = (S2+S3)/2, M3=(S3+S4)/2, M4=(S4+S1)/2; pair Sp1 = (7.5, 7.5), Sp2=(-7.5, 7.5), Sp3 = (-7.5, -7.5), Sp4 = (7.5, -7.5); draw(S1--S2--S3--S4--cycle); draw(Sp1--Sp2--Sp3--Sp4--cycle); draw(Sp1--M1--Sp2--M2--Sp3--M3--Sp4--M4--cycle); [/asy]\n",
      "\n",
      "0.9002006649971008\n",
      "A game uses a deck of $n$ different cards, where $n$ is an integer and $n \\geq 6.$ The number of possible sets of 6 cards that can be drawn from the deck is 6 times the number of possible sets of 3 cards that can be drawn. Find $n.$\n",
      "Found similar problem: A game uses a deck of $n$ different cards, where $n$ is an integer and $n \\geq 6.$ The number of possible sets of 6 cards that can be drawn from the deck is 6 times the number of possible sets of 3 cards that can be drawn. Find $n.$\n",
      " ~ pi_is_3.14\n",
      "\n",
      "1.0000001192092896\n",
      "Find the number of five-digit positive integers, $n$, that satisfy the following conditions:\n",
      "\n",
      "\n",
      "(a) the number $n$ is divisible by $5,$\n",
      "\n",
      "\n",
      "(b) the first and last digits of $n$ are equal, and\n",
      "\n",
      "\n",
      "(c) the sum of the digits of $n$ is divisible by $5.$\n",
      "Found similar problem: Find the number of five-digit positive integers, $n$, that satisfy the following conditions:\n",
      " (a) the number $n$ is divisible by $5,$\n",
      " (b) the first and last digits of $n$ are equal, and\n",
      " (c) the sum of the digits of $n$ is divisible by $5.$\n",
      "\n",
      "1.0000001192092896\n",
      "Circles $\\omega_1$ and $\\omega_2$ with radii $961$ and $625$, respectively, intersect at distinct points $A$ and $B$. A third circle $\\omega$ is externally tangent to both $\\omega_1$ and $\\omega_2$. Suppose line $AB$ intersects $\\omega$ at two points $P$ and $Q$ such that the measure of minor arc $\\widehat{PQ}$ is $120^{\\circ}$. Find the distance between the centers of $\\omega_1$ and $\\omega_2$.\n",
      "Found similar problem: Circles $\\omega_1$ and $\\omega_2$ with radii $961$ and $625$, respectively, intersect at distinct points $A$ and $B$. A third circle $\\omega$ is externally tangent to both $\\omega_1$ and $\\omega_2$. Suppose line $AB$ intersects $\\omega$ at two points $P$ and $Q$ such that the measure of minor arc $\\widehat{PQ}$ is $120^{\\circ}$. Find the distance between the centers of $\\omega_1$ and $\\omega_2$.\n",
      "\n",
      "1.0\n",
      "Freddy the frog is jumping around the coordinate plane searching for a river, which lies on the horizontal line $y = 24$. A fence is located at the horizontal line $y = 0$. On each jump Freddy randomly chooses a direction parallel to one of the coordinate axes and moves one unit in that direction. When he is at a point where $y=0$, with equal likelihoods he chooses one of three directions where he either jumps parallel to the fence or jumps away from the fence, but he never chooses the direction that would have him cross over the fence to where $y < 0$. Freddy starts his search at the point $(0, 21)$ and will stop once he reaches a point on the river. Find the expected number of jumps it will take Freddy to reach the river.\n",
      "Found similar problem: Freddy the frog is jumping around the coordinate plane searching for a river, which lies on the horizontal line $y = 24$. A fence is located at the horizontal line $y = 0$. On each jump Freddy randomly chooses a direction parallel to one of the coordinate axes and moves one unit in that direction. When he is at a point where $y=0$, with equal likelihoods he chooses one of three directions where he either jumps parallel to the fence or jumps away from the fence, but he never chooses the direction that would have him cross over the fence to where $y < 0$. Freddy starts his search at the point $(0, 21)$ and will stop once he reaches a point on the river. Find the expected number of jumps it will take Freddy to reach the river.\n",
      "\n",
      "1.0\n",
      "While watching a show, Ayako, Billy, Carlos, Dahlia, Ehuang, and Frank sat in that order in a row of six chairs. During the break, they went to the kitchen for a snack. When they came back, they sat on those six chairs in such a way that if two of them sat next to each other before the break, then they did not sit next to each other after the break. Find the number of possible seating orders they could have chosen after the break.\n",
      "Found similar problem: While watching a show, Ayako, Billy, Carlos, Dahlia, Ehuang, and Frank sat in that order in a row of six chairs. During the break, they went to the kitchen for a snack. When they came back, they sat on those six chairs in such a way that if two of them sat next to each other before the break, then they did not sit next to each other after the break. Find the number of possible seating orders they could have chosen after the break.\n",
      "\n",
      "0.6990405321121216\n",
      "Triangle $ABC$ is an isosceles right triangle with $AB=AC=3$. Let $M$ be the midpoint of hypotenuse $\\overline{BC}$. Points $I$ and $E$ lie on sides $\\overline{AC}$ and $\\overline{AB}$, respectively, so that $AI>AE$ and $AIME$ is a cyclic quadrilateral. Given that triangle $EMI$ has area $2$, the length $CI$ can be written as $\\frac{a-\\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers and $b$ is not divisible by the square of any prime. What is the value of $a+b+c$?\n",
      "$\\textbf{(A) }9 \\qquad \\textbf{(B) }10 \\qquad \\textbf{(C) }11 \\qquad \\textbf{(D) }12 \\qquad \\textbf{(E) }13 \\qquad$\n",
      "Found similar problem: In $\\triangle ABC$, $AC = BC$, and point $D$ is on $\\overline{BC}$ so that $CD = 3\\cdot BD$. Let $E$ be the midpoint of $\\overline{AD}$. Given that $CE = \\sqrt{7}$ and $BE = 3$, the area of $\\triangle ABC$ can be expressed in the form $m\\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m+n$.\n",
      "\n",
      "0.9835737347602844\n",
      "Let $ABCD$ be a [parallelogram](https://artofproblemsolving.com/wiki/index.php/Parallelogram).  Extend $\\overline{DA}$ through $A$ to a point $P,$ and let $\\overline{PC}$ meet $\\overline{AB}$ at $Q$ and $\\overline{DB}$ at $R.$  Given that $PQ = 735$ and $QR = 112,$ find $RC.$\n",
      "Found similar problem: Let $ABCD$ be a parallelogram. Extend $\\overline{DA}$ through $A$ to a point $P,$ and let $\\overline{PC}$ meet $\\overline{AB}$ at $Q$ and $\\overline{DB}$ at $R.$ Given that $PQ = 735$ and $QR = 112,$ find $RC.$\n",
      "\n",
      "1.0000001192092896\n",
      "Let $S$ be the set of all polynomials of the form $z^3 + az^2 + bz + c$, where $a$, $b$, and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either $|z| = 20$ or $|z| = 13$.\n",
      "Found similar problem: Let $S$ be the set of all polynomials of the form $z^3 + az^2 + bz + c$, where $a$, $b$, and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either $|z| = 20$ or $|z| = 13$.\n",
      "\n",
      "1.0\n",
      "A rational number written in base eight is $\\underline{ab} . \\underline{cd}$, where all digits are nonzero. The same number in base twelve is $\\underline{bb} . \\underline{ba}$. Find the base-ten number $\\underline{abc}$.\n",
      "Found similar problem: A rational number written in base eight is $\\underline{ab} . \\underline{cd}$, where all digits are nonzero. The same number in base twelve is $\\underline{bb} . \\underline{ba}$. Find the base-ten number $\\underline{abc}$.\n",
      "\n",
      "1.0000001192092896\n",
      "What is the largest even integer that cannot be written as the sum of two odd composite numbers?\n",
      "Found similar problem: What is the largest even integer that cannot be written as the sum of two odd composite numbers?\n",
      "\n",
      "1.0\n",
      "Call a three-term strictly increasing arithmetic sequence of integers special if the sum of the squares of the three terms equals the product of the middle term and the square of the common difference. Find the sum of the third terms of all special sequences.\n",
      "Found similar problem: Call a three-term strictly increasing arithmetic sequence of integers special if the sum of the squares of the three terms equals the product of the middle term and the square of the common difference. Find the sum of the third terms of all special sequences.\n",
      "\n",
      "1.0\n",
      "The polynomial $f(z)=az^{2018}+bz^{2017}+cz^{2016}$ has real coefficients not exceeding $2019$, and $f\\left(\\tfrac{1+\\sqrt{3}i}{2}\\right)=2015+2019\\sqrt{3}i$. Find the remainder when $f(1)$ is divided by $1000$.\n",
      "Found similar problem: The polynomial $f(z)=az^{2018}+bz^{2017}+cz^{2016}$ has real coefficients not exceeding $2019$, and $f\\left(\\tfrac{1+\\sqrt{3}i}{2}\\right)=2015+2019\\sqrt{3}i$. Find the remainder when $f(1)$ is divided by $1000$.\n",
      "\n",
      "1.0\n",
      "A convex quadrilateral has area $30$ and side lengths $5, 6, 9,$ and $7,$ in that order. Denote by $\\theta$ the measure of the acute angle formed by the diagonals of the quadrilateral. Then $\\tan \\theta$ can be written in the form $\\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.\n",
      "Found similar problem: A convex quadrilateral has area $30$ and side lengths $5, 6, 9,$ and $7,$ in that order. Denote by $\\theta$ the measure of the acute angle formed by the diagonals of the quadrilateral. Then $\\tan \\theta$ can be written in the form $\\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.\n",
      "\n",
      "1.0\n",
      "Find the positive integer $n\\,$ for which\n",
      "\\[\\lfloor\\log_2{1}\\rfloor+\\lfloor\\log_2{2}\\rfloor+\\lfloor\\log_2{3}\\rfloor+\\cdots+\\lfloor\\log_2{n}\\rfloor=1994\\]\n",
      "(For real $x\\,$, $\\lfloor x\\rfloor\\,$ is the greatest integer $\\le x.\\,$)\n",
      "Found similar problem: Find the positive integer $n\\,$ for which \\[\\lfloor\\log_2{1}\\rfloor+\\lfloor\\log_2{2}\\rfloor+\\lfloor\\log_2{3}\\rfloor+\\cdots+\\lfloor\\log_2{n}\\rfloor=1994\\] (For real $x\\,$, $\\lfloor x\\rfloor\\,$ is the greatest integer $\\le x.\\,$)\n",
      "\n",
      "1.0\n",
      "Rectangle $ABCD$ has side lengths $AB=84$ and $AD=42$. Point $M$ is the midpoint of $\\overline{AD}$, point $N$ is the trisection point of $\\overline{AB}$ closer to $A$, and point $O$ is the intersection of $\\overline{CM}$ and $\\overline{DN}$. Point $P$ lies on the quadrilateral $BCON$, and $\\overline{BP}$ bisects the area of $BCON$. Find the area of $\\triangle CDP$.\n",
      "Found similar problem: Rectangle $ABCD$ has side lengths $AB=84$ and $AD=42$. Point $M$ is the midpoint of $\\overline{AD}$, point $N$ is the trisection point of $\\overline{AB}$ closer to $A$, and point $O$ is the intersection of $\\overline{CM}$ and $\\overline{DN}$. Point $P$ lies on the quadrilateral $BCON$, and $\\overline{BP}$ bisects the area of $BCON$. Find the area of $\\triangle CDP$.\n",
      "\n",
      "1.0\n",
      "Find the sum of all positive integers $n$ such that $\\sqrt{n^2+85n+2017}$ is an integer.\n",
      "Found similar problem: Find the sum of all positive integers $n$ such that $\\sqrt{n^2+85n+2017}$ is an integer.\n",
      "\n",
      "0.9702553153038025\n",
      "A beam of light strikes $\\overline{BC}\\,$ at point $C\\,$ with angle of incidence $\\alpha=19.94^\\circ\\,$ and reflects with an equal angle of reflection as shown.  The light beam continues its path, reflecting off line segments $\\overline{AB}\\,$ and $\\overline{BC}\\,$ according to the rule: angle of incidence equals angle of reflection.  Given that $\\beta=\\alpha/10=1.994^\\circ\\,$ and $AB=BC,\\,$ determine the number of times the light beam will bounce off the two line segments.  Include the first reflection at $C\\,$ in your count.\n",
      "[AIME 1994 Problem 14.png](https://artofproblemsolving.com/wiki/index.php/File:AIME_1994_Problem_14.png)\n",
      "Found similar problem: A beam of light strikes $\\overline{BC}\\,$ at point $C\\,$ with angle of incidence $\\alpha=19.94^\\circ\\,$ and reflects with an equal angle of reflection as shown. The light beam continues its path, reflecting off line segments $\\overline{AB}\\,$ and $\\overline{BC}\\,$ according to the rule: angle of incidence equals angle of reflection. Given that $\\beta=\\alpha/10=1.994^\\circ\\,$ and $AB=BC,\\,$ determine the number of times the light beam will bounce off the two line segments. Include the first reflection at $C\\,$ in your count.\n",
      "\n",
      "1.0\n",
      "Let $a, b, c,$ and $d$ be real numbers that satisfy the system of equations\n",
      "\\begin{align*} a + b &= -3, \\\\ ab + bc + ca &= -4, \\\\ abc + bcd + cda + dab &= 14, \\\\ abcd &= 30. \\end{align*}\n",
      "There exist relatively prime positive integers $m$ and $n$ such that\n",
      "\\[a^2 + b^2 + c^2 + d^2 = \\frac{m}{n}.\\]Find $m + n$.\n",
      "Found similar problem: Let $a, b, c,$ and $d$ be real numbers that satisfy the system of equations \\begin{align*} a + b &= -3, \\\\ ab + bc + ca &= -4, \\\\ abc + bcd + cda + dab &= 14, \\\\ abcd &= 30. \\end{align*} There exist relatively prime positive integers $m$ and $n$ such that \\[a^2 + b^2 + c^2 + d^2 = \\frac{m}{n}.\\]Find $m + n$.\n",
      "\n",
      "1.0\n",
      "Let $L$ be the line with slope $\\frac{5}{12}$ that contains the point $A=(24,-1)$, and let $M$ be the line perpendicular to line $L$ that contains the point $B=(5,6)$.  The original coordinate axes are erased, and line $L$ is made the $x$-axis and line $M$ the $y$-axis.  In the new coordinate system, point $A$ is on the positive $x$-axis, and point $B$ is on the positive $y$-axis.  The point $P$ with coordinates $(-14,27)$ in the original system has coordinates $(\\alpha,\\beta)$ in the new coordinate system.  Find $\\alpha+\\beta$.\n",
      "Found similar problem: Let $L$ be the line with slope $\\frac{5}{12}$ that contains the point $A=(24,-1)$, and let $M$ be the line perpendicular to line $L$ that contains the point $B=(5,6)$. The original coordinate axes are erased, and line $L$ is made the $x$-axis and line $M$ the $y$-axis. In the new coordinate system, point $A$ is on the positive $x$-axis, and point $B$ is on the positive $y$-axis. The point $P$ with coordinates $(-14,27)$ in the original system has coordinates $(\\alpha,\\beta)$ in the new coordinate system. Find $\\alpha+\\beta$.\n",
      "\n",
      "1.0\n",
      "Zou and Chou are practicing their $100$-meter sprints by running $6$ races against each other. Zou wins the first race, and after that, the probability that one of them wins a race is $\\frac23$ if they won the previous race but only $\\frac13$ if they lost the previous race. The probability that Zou will win exactly $5$ of the $6$ races is $\\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$\n",
      "Found similar problem: Zou and Chou are practicing their $100$-meter sprints by running $6$ races against each other. Zou wins the first race, and after that, the probability that one of them wins a race is $\\frac23$ if they won the previous race but only $\\frac13$ if they lost the previous race. The probability that Zou will win exactly $5$ of the $6$ races is $\\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$\n",
      "\n",
      "1.0000001192092896\n",
      "Ms. Math's kindergarten class has $16$ registered students. The classroom has a very large number, $N$, of play blocks which satisfies the conditions:\n",
      "(a) If $16$, $15$, or $14$ students are present in the class, then in each case all the blocks can be distributed in equal numbers to each student, and\n",
      "(b) There are three integers $0 < x < y < z < 14$ such that when $x$, $y$, or $z$ students are present and the blocks are distributed in equal numbers to each student, there are exactly three blocks left over.\n",
      "Find the sum of the distinct prime divisors of the least possible value of $N$ satisfying the above conditions.\n",
      "Found similar problem: Ms. Math's kindergarten class has $16$ registered students. The classroom has a very large number, $N$, of play blocks which satisfies the conditions:\n",
      " (a) If $16$, $15$, or $14$ students are present in the class, then in each case all the blocks can be distributed in equal numbers to each student, and\n",
      " (b) There are three integers $0 < x < y < z < 14$ such that when $x$, $y$, or $z$ students are present and the blocks are distributed in equal numbers to each student, there are exactly three blocks left over.\n",
      " Find the sum of the distinct prime divisors of the least possible value of $N$ satisfying the above conditions.\n",
      "\n",
      "1.0\n",
      "A rectangle has sides of length $a$ and 36. A hinge is installed at each vertex of the rectangle, and at the midpoint of each side of length 36. The sides of length $a$ can be pressed toward each other keeping those two sides parallel so the rectangle becomes a convex hexagon as shown. When the figure is a hexagon with the sides of length $a$ parallel and separated by a distance of 24, the hexagon has the same area as the original rectangle. Find $a^2$. \n",
      "[asy] pair A,B,C,D,E,F,R,S,T,X,Y,Z; dotfactor = 2; unitsize(.1cm); A = (0,0); B = (0,18); C = (0,36); // don't look here D = (12*2.236, 36); E = (12*2.236, 18); F = (12*2.236, 0); draw(A--B--C--D--E--F--cycle); dot(\" \",A,NW); dot(\" \",B,NW); dot(\" \",C,NW); dot(\" \",D,NW); dot(\" \",E,NW); dot(\" \",F,NW); //don't look here R = (12*2.236 +22,0); S = (12*2.236 + 22 - 13.4164,12); T = (12*2.236 + 22,24); X = (12*4.472+ 22,24); Y = (12*4.472+ 22 + 13.4164,12); Z = (12*4.472+ 22,0); draw(R--S--T--X--Y--Z--cycle); dot(\" \",R,NW); dot(\" \",S,NW); dot(\" \",T,NW); dot(\" \",X,NW); dot(\" \",Y,NW); dot(\" \",Z,NW); // sqrt180 = 13.4164 // sqrt5 = 2.236[/asy]\n",
      "Found similar problem: A rectangle has sides of length $a$ and 36. A hinge is installed at each vertex of the rectangle, and at the midpoint of each side of length 36. The sides of length $a$ can be pressed toward each other keeping those two sides parallel so the rectangle becomes a convex hexagon as shown. When the figure is a hexagon with the sides of length $a$ parallel and separated by a distance of 24, the hexagon has the same area as the original rectangle. Find $a^2$.\n",
      "[asy] pair A,B,C,D,E,F,R,S,T,X,Y,Z; dotfactor = 2; unitsize(.1cm); A = (0,0); B = (0,18); C = (0,36); // don't look here D = (12*2.236, 36); E = (12*2.236, 18); F = (12*2.236, 0); draw(A--B--C--D--E--F--cycle); dot(\" \",A,NW); dot(\" \",B,NW); dot(\" \",C,NW); dot(\" \",D,NW); dot(\" \",E,NW); dot(\" \",F,NW); //don't look here R = (12*2.236 +22,0); S = (12*2.236 + 22 - 13.4164,12); T = (12*2.236 + 22,24); X = (12*4.472+ 22,24); Y = (12*4.472+ 22 + 13.4164,12); Z = (12*4.472+ 22,0); draw(R--S--T--X--Y--Z--cycle); dot(\" \",R,NW); dot(\" \",S,NW); dot(\" \",T,NW); dot(\" \",X,NW); dot(\" \",Y,NW); dot(\" \",Z,NW); // sqrt180 = 13.4164 // sqrt5 = 2.236[/asy]\n",
      "\n",
      "1.0000001192092896\n",
      "A regular icosahedron is a $20$-faced solid where each face is an equilateral triangle and five triangles meet at every vertex. The regular icosahedron shown below has one vertex at the top, one vertex at the bottom, an upper pentagon of five vertices all adjacent to the top vertex and all in the same horizontal plane, and a lower pentagon of five vertices all adjacent to the bottom vertex and all in another horizontal plane. Find the number of paths from the top vertex to the bottom vertex such that each part of a path goes downward or horizontally along an edge of the icosahedron, and no vertex is repeated.\n",
      "[asy] size(3cm); pair A=(0.05,0),B=(-.9,-0.6),C=(0,-0.45),D=(.9,-0.6),E=(.55,-0.85),F=(-0.55,-0.85),G=B-(0,1.1),H=F-(0,0.6),I=E-(0,0.6),J=D-(0,1.1),K=C-(0,1.4),L=C+K-A; draw(A--B--F--E--D--A--E--A--F--A^^B--G--F--K--G--L--J--K--E--J--D--J--L--K); draw(B--C--D--C--A--C--H--I--C--H--G^^H--L--I--J^^I--D^^H--B,dashed); dot(A^^B^^C^^D^^E^^F^^G^^H^^I^^J^^K^^L); [/asy]\n",
      "Found similar problem: A regular icosahedron is a $20$-faced solid where each face is an equilateral triangle and five triangles meet at every vertex. The regular icosahedron shown below has one vertex at the top, one vertex at the bottom, an upper pentagon of five vertices all adjacent to the top vertex and all in the same horizontal plane, and a lower pentagon of five vertices all adjacent to the bottom vertex and all in another horizontal plane. Find the number of paths from the top vertex to the bottom vertex such that each part of a path goes downward or horizontally along an edge of the icosahedron, and no vertex is repeated.\n",
      "[asy] size(3cm); pair A=(0.05,0),B=(-.9,-0.6),C=(0,-0.45),D=(.9,-0.6),E=(.55,-0.85),F=(-0.55,-0.85),G=B-(0,1.1),H=F-(0,0.6),I=E-(0,0.6),J=D-(0,1.1),K=C-(0,1.4),L=C+K-A; draw(A--B--F--E--D--A--E--A--F--A^^B--G--F--K--G--L--J--K--E--J--D--J--L--K); draw(B--C--D--C--A--C--H--I--C--H--G^^H--L--I--J^^I--D^^H--B,dashed); dot(A^^B^^C^^D^^E^^F^^G^^H^^I^^J^^K^^L); [/asy]\n",
      "\n",
      "1.0\n",
      "A long thin strip of paper is $1024$ units in length, $1$ unit in width, and is divided into $1024$ unit squares. The paper is folded in half repeatedly. For the first fold, the right end of the paper is folded over to coincide with and lie on top of the left end. The result is a $512$ by $1$ strip of double thickness. Next, the right end of this strip is folded over to coincide with and lie on top of the left end, resulting in a $256$ by $1$ strip of quadruple thickness. This process is repeated $8$ more times. After the last fold, the strip has become a stack of $1024$ unit squares. How many of these squares lie below the square that was originally the $942$nd square counting from the left?\n",
      "Found similar problem: A long thin strip of paper is $1024$ units in length, $1$ unit in width, and is divided into $1024$ unit squares. The paper is folded in half repeatedly. For the first fold, the right end of the paper is folded over to coincide with and lie on top of the left end. The result is a $512$ by $1$ strip of double thickness. Next, the right end of this strip is folded over to coincide with and lie on top of the left end, resulting in a $256$ by $1$ strip of quadruple thickness. This process is repeated $8$ more times. After the last fold, the strip has become a stack of $1024$ unit squares. How many of these squares lie below the square that was originally the $942$nd square counting from the left?\n",
      "\n",
      "0.9605407118797302\n",
      "The increasing [geometric sequence](https://artofproblemsolving.com/wiki/index.php/Geometric_sequence) $x_{0},x_{1},x_{2},\\ldots$ consists entirely of [integral](https://artofproblemsolving.com/wiki/index.php/Integer) powers of $3.$ Given that\n",
      "$\\sum_{n=0}^{7}\\log_{3}(x_{n}) = 308$ and $56 \\leq \\log_{3}\\left ( \\sum_{n=0}^{7}x_{n}\\right ) \\leq 57,$\n",
      "find $\\log_{3}(x_{14}).$\n",
      "Found similar problem: The increasing geometric sequence $x_{0},x_{1},x_{2},\\ldots$ consists entirely of integral powers of $3.$ Given that\n",
      "$\\sum_{n=0}^{7}\\log_{3}(x_{n}) = 308$ and $56 \\leq \\log_{3}\\left ( \\sum_{n=0}^{7}x_{n}\\right ) \\leq 57,$\n",
      " find $\\log_{3}(x_{14}).$\n",
      "\n",
      "0.9999999403953552\n",
      "Let $a > 1$ and $x > 1$ satisfy $\\log_a(\\log_a(\\log_a 2) + \\log_a 24 - 128) = 128$ and $\\log_a(\\log_a x) = 256$. Find the remainder when $x$ is divided by $1000$.\n",
      "Found similar problem: Let $a > 1$ and $x > 1$ satisfy $\\log_a(\\log_a(\\log_a 2) + \\log_a 24 - 128) = 128$ and $\\log_a(\\log_a x) = 256$. Find the remainder when $x$ is divided by $1000$.\n",
      "\n",
      "1.0\n",
      "Let $A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8$ be a regular octagon.  Let $M_1$, $M_3$, $M_5$, and $M_7$ be the midpoints of sides $\\overline{A_1 A_2}$, $\\overline{A_3 A_4}$, $\\overline{A_5 A_6}$, and $\\overline{A_7 A_8}$, respectively.  For $i = 1, 3, 5, 7$, ray $R_i$ is constructed from $M_i$ towards the interior of the octagon such that $R_1 \\perp R_3$, $R_3 \\perp R_5$, $R_5 \\perp R_7$, and $R_7 \\perp R_1$.  Pairs of rays $R_1$ and $R_3$, $R_3$ and $R_5$, $R_5$ and $R_7$, and $R_7$ and $R_1$ meet at $B_1$, $B_3$, $B_5$, $B_7$ respectively.  If $B_1 B_3 = A_1 A_2$, then $\\cos 2 \\angle A_3 M_3 B_1$ can be written in the form $m - \\sqrt{n}$, where $m$ and $n$ are positive integers.  Find $m + n$.\n",
      "Found similar problem: Let $A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8$ be a regular octagon. Let $M_1$, $M_3$, $M_5$, and $M_7$ be the midpoints of sides $\\overline{A_1 A_2}$, $\\overline{A_3 A_4}$, $\\overline{A_5 A_6}$, and $\\overline{A_7 A_8}$, respectively. For $i = 1, 3, 5, 7$, ray $R_i$ is constructed from $M_i$ towards the interior of the octagon such that $R_1 \\perp R_3$, $R_3 \\perp R_5$, $R_5 \\perp R_7$, and $R_7 \\perp R_1$. Pairs of rays $R_1$ and $R_3$, $R_3$ and $R_5$, $R_5$ and $R_7$, and $R_7$ and $R_1$ meet at $B_1$, $B_3$, $B_5$, $B_7$ respectively. If $B_1 B_3 = A_1 A_2$, then $\\cos 2 \\angle A_3 M_3 B_1$ can be written in the form $m - \\sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.\n",
      "Diagram\n",
      "[asy] size(250); pair A,B,C,D,E,F,G,H,M,N,O,O2,P,W,X,Y,Z; A=(-76.537,184.776); B=(76.537,184.776); C=(184.776,76.537); D=(184.776,-76.537); E=(76.537,-184.776); F=(-76.537,-184.776); G=(-184.776,-76.537); H=(-184.776,76.537); M=(A+B)/2; N=(C+D)/2; O=(E+F)/2; O2=(A+E)/2; P=(G+H)/2; W=(100,-41.421); X=(-41.421,-100); Y=(-100,41.421); Z=(41.421,100); draw(A--B--C--D--E--F--G--H--A); label(\"$A_1$\",A,dir(112.5)); label(\"$A_2$\",B,dir(67.5)); label(\"$\\textcolor{blue}{A_3}$\",C,dir(22.5)); label(\"$A_4$\",D,dir(337.5)); label(\"$A_5$\",E,dir(292.5)); label(\"$A_6$\",F,dir(247.5)); label(\"$A_7$\",G,dir(202.5)); label(\"$A_8$\",H,dir(152.5)); label(\"$M_1$\",M,dir(90)); label(\"$\\textcolor{blue}{M_3}$\",N,dir(0)); label(\"$M_5$\",O,dir(270)); label(\"$M_7$\",P,dir(180)); label(\"$O$\",O2,dir(152.5)); draw(M--W,red); draw(N--X,red); draw(O--Y,red); draw(P--Z,red); draw(O2--(W+X)/2,red); draw(O2--N,red); label(\"$\\textcolor{blue}{B_1}$\",W,dir(292.5)); label(\"$B_2$\",(W+X)/2,dir(292.5)); label(\"$B_3$\",X,dir(202.5)); label(\"$B_5$\",Y,dir(112.5)); label(\"$B_7$\",Z,dir(22.5)); [/asy] All distances are to scale.\n",
      "\n",
      "1.0\n",
      "There are $n$ mathematicians seated around a circular table with $n$ seats numbered $1,$ $2,$ $3,$ $...,$ $n$ in clockwise order. After a break they again sit around the table. The mathematicians note that there is a positive integer $a$ such that\n",
      "\n",
      "\n",
      "($1$) for each $k,$ the mathematician who was seated in seat $k$ before the break is seated in seat $ka$ after the break (where seat $i + n$ is seat $i$);\n",
      "\n",
      "\n",
      "($2$) for every pair of mathematicians, the number of mathematicians sitting between them after the break, counting in both the clockwise and the counterclockwise directions, is different from either of the number of mathematicians sitting between them before the break.\n",
      "\n",
      "Find the number of possible values of $n$ with $1 < n < 1000.$\n",
      "Found similar problem: There are $n$ mathematicians seated around a circular table with $n$ seats numbered $1,$ $2,$ $3,$ $...,$ $n$ in clockwise order. After a break they again sit around the table. The mathematicians note that there is a positive integer $a$ such that\n",
      " ($1$) for each $k,$ the mathematician who was seated in seat $k$ before the break is seated in seat $ka$ after the break (where seat $i + n$ is seat $i$);\n",
      " ($2$) for every pair of mathematicians, the number of mathematicians sitting between them after the break, counting in both the clockwise and the counterclockwise directions, is different from either of the number of mathematicians sitting between them before the break.\n",
      " Find the number of possible values of $n$ with $1 < n < 1000.$\n",
      "\n",
      "1.0000001192092896\n",
      "Given that $A_k = \\frac {k(k - 1)}2\\cos\\frac {k(k - 1)\\pi}2,$ find $|A_{19} + A_{20} + \\cdots + A_{98}|.$\n",
      "Found similar problem: Given that $A_k = \\frac {k(k - 1)}2\\cos\\frac {k(k - 1)\\pi}2,$ find $|A_{19} + A_{20} + \\cdots + A_{98}|.$\n",
      "\n",
      "1.0\n",
      "Circle $C_0$ has radius $1$, and the point $A_0$ is a point on the circle. Circle $C_1$ has radius $r<1$ and is internally tangent to $C_0$ at point $A_0$. Point $A_1$ lies on circle $C_1$ so that $A_1$ is located $90^{\\circ}$ counterclockwise from $A_0$ on $C_1$. Circle $C_2$ has radius $r^2$ and is internally tangent to $C_1$ at point $A_1$. In this way a sequence of circles $C_1,C_2,C_3,\\ldots$ and a sequence of points on the circles $A_1,A_2,A_3,\\ldots$ are constructed, where circle $C_n$ has radius $r^n$ and is internally tangent to circle $C_{n-1}$ at point $A_{n-1}$, and point $A_n$ lies on $C_n$ $90^{\\circ}$ counterclockwise from point $A_{n-1}$, as shown in the figure below. There is one point $B$ inside all of these circles. When $r = \\frac{11}{60}$, the distance from the center $C_0$ to $B$ is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n",
      "Found similar problem: Circle $C_0$ has radius $1$, and the point $A_0$ is a point on the circle. Circle $C_1$ has radius $r<1$ and is internally tangent to $C_0$ at point $A_0$. Point $A_1$ lies on circle $C_1$ so that $A_1$ is located $90^{\\circ}$ counterclockwise from $A_0$ on $C_1$. Circle $C_2$ has radius $r^2$ and is internally tangent to $C_1$ at point $A_1$. In this way a sequence of circles $C_1,C_2,C_3,\\ldots$ and a sequence of points on the circles $A_1,A_2,A_3,\\ldots$ are constructed, where circle $C_n$ has radius $r^n$ and is internally tangent to circle $C_{n-1}$ at point $A_{n-1}$, and point $A_n$ lies on $C_n$ $90^{\\circ}$ counterclockwise from point $A_{n-1}$, as shown in the figure below. There is one point $B$ inside all of these circles. When $r = \\frac{11}{60}$, the distance from the center $C_0$ to $B$ is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n",
      "[asy] draw(Circle((0,0),125)); draw(Circle((25,0),100)); draw(Circle((25,20),80)); draw(Circle((9,20),64)); dot((125,0)); label(\"$A_0$\",(125,0),E); dot((25,100)); label(\"$A_1$\",(25,100),SE); dot((-55,20)); label(\"$A_2$\",(-55,20),E); [/asy]\n",
      "\n",
      "1.0\n",
      "A disk with radius $1$ is externally tangent to a disk with radius $5$. Let $A$ be the point where the disks are tangent, $C$ be the center of the smaller disk, and $E$ be the center of the larger disk. While the larger disk remains fixed, the smaller disk is allowed to roll along the outside of the larger disk until the smaller disk has turned through an angle of $360^\\circ$. That is, if the center of the smaller disk has moved to the point $D$, and the point on the smaller disk that began at $A$ has now moved to point $B$, then $\\overline{AC}$ is parallel to $\\overline{BD}$. Then $\\sin^2(\\angle BEA)=\\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n",
      "Found similar problem: A disk with radius $1$ is externally tangent to a disk with radius $5$. Let $A$ be the point where the disks are tangent, $C$ be the center of the smaller disk, and $E$ be the center of the larger disk. While the larger disk remains fixed, the smaller disk is allowed to roll along the outside of the larger disk until the smaller disk has turned through an angle of $360^\\circ$. That is, if the center of the smaller disk has moved to the point $D$, and the point on the smaller disk that began at $A$ has now moved to point $B$, then $\\overline{AC}$ is parallel to $\\overline{BD}$. Then $\\sin^2(\\angle BEA)=\\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n",
      "\n",
      "1.0\n",
      "Let $N$ be the least positive integer that is both $22$ percent less than one integer and $16$ percent greater than another integer. Find the remainder when $N$ is divided by $1000$.\n",
      "Found similar problem: Let $N$ be the least positive integer that is both $22$ percent less than one integer and $16$ percent greater than another integer. Find the remainder when $N$ is divided by $1000$.\n",
      "\n",
      "1.0\n",
      "For every subset $T$ of $U = \\{ 1,2,3,\\ldots,18 \\}$, let $s(T)$ be the sum of the elements of $T$, with $s(\\emptyset)$ defined to be $0$. If $T$ is chosen at random among all subsets of $U$, the probability that $s(T)$ is divisible by $3$ is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m$.\n",
      "Found similar problem: For every subset $T$ of $U = \\{ 1,2,3,\\ldots,18 \\}$, let $s(T)$ be the sum of the elements of $T$, with $s(\\emptyset)$ defined to be $0$. If $T$ is chosen at random among all subsets of $U$, the probability that $s(T)$ is divisible by $3$ is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m$.\n",
      "\n",
      "0.9650662541389465\n",
      "Point $P$ is inside $\\triangle ABC$. Line segments $APD$, $BPE$, and $CPF$ are drawn with $D$ on $BC$, $E$ on $AC$, and $F$ on $AB$ (see the figure below). Given that $AP=6$, $BP=9$, $PD=6$, $PE=3$, and $CF=20$, find the area of $\\triangle ABC$.\n",
      "\n",
      "[AIME 1989 Problem 15.png](https://artofproblemsolving.com/wiki/index.php/File:AIME_1989_Problem_15.png)\n",
      "Found similar problem: Point $P$ is inside $\\triangle ABC$. Line segments $APD$, $BPE$, and $CPF$ are drawn with $D$ on $BC$, $E$ on $AC$, and $F$ on $AB$ (see the figure below). Given that $AP=6$, $BP=9$, $PD=6$, $PE=3$, and $CF=20$, find the area of $\\triangle ABC$.\n",
      "\n",
      "0.9581680297851562\n",
      "Four lighthouses are located at points $A$, $B$, $C$, and $D$. The lighthouse at $A$ is $5$ kilometers from the lighthouse at $B$, the lighthouse at $B$ is $12$ kilometers from the lighthouse at $C$, and the lighthouse at $A$ is $13$ kilometers from the lighthouse at $C$. To an observer at $A$, the angle determined by the lights at $B$ and $D$ and the angle determined by the lights at $C$ and $D$ are equal. To an observer at $C$, the angle determined by the lights at $A$ and $B$ and the angle determined by the lights at $D$ and $B$ are equal. The number of kilometers from $A$ to $D$ is given by $\\frac {p\\sqrt{q}}{r}$, where $p$, $q$, and $r$ are relatively prime positive integers, and $r$ is not divisible by the square of any prime. Find $p$ + $q$ + $r$.\n",
      "Found similar problem: Four lighthouses are located at points $A$, $B$, $C$, and $D$. The lighthouse at $A$ is $5$ kilometers from the lighthouse at $B$, the lighthouse at $B$ is $12$ kilometers from the lighthouse at $C$, and the lighthouse at $A$ is $13$ kilometers from the lighthouse at $C$. To an observer at $A$, the angle determined by the lights at $B$ and $D$ and the angle determined by the lights at $C$ and $D$ are equal. To an observer at $C$, the angle determined by the lights at $A$ and $B$ and the angle determined by the lights at $D$ and $B$ are equal. The number of kilometers from $A$ to $D$ is given by $\\frac {p\\sqrt{q}}{r}$, where $p$, $q$, and $r$ are relatively prime positive integers, and $r$ is not divisible by the square of any prime. Find $p$ + $q$ + $r$.\n",
      "Diagram\n",
      "[asy] size(120); pathpen = linewidth(0.7); pointpen = black; pen f = fontsize(10); pair B=(0,0), A=(5,0), C=(0,13), E=(-5,0), O = incenter(E,C,A), D=IP(A -- A+3*(O-A),E--C); D(A--B--C--cycle); D(A--D--C); D(D--E--B, linetype(\"4 4\")); MP(\"5\", (A+B)/2, f); MP(\"13\", (A+C)/2, NE,f); MP(\"A\",D(A),f); MP(\"B\",D(B),f); MP(\"C\",D(C),N,f); MP(\"A'\",D(E),f); MP(\"D\",D(D),NW,f); D(rightanglemark(C,B,A,20)); D(anglemark(D,A,E,35));D(anglemark(C,A,D,30)); [/asy] -asjpz\n",
      "\n",
      "1.0000001192092896\n",
      "Let $A=\\{1,2,3,4\\}$, and $f$ and $g$ be randomly chosen (not necessarily distinct) functions from $A$ to $A$. The probability that the range of $f$ and the range of $g$ are disjoint is $\\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m$.\n",
      "Found similar problem: Let $A=\\{1,2,3,4\\}$, and $f$ and $g$ be randomly chosen (not necessarily distinct) functions from $A$ to $A$. The probability that the range of $f$ and the range of $g$ are disjoint is $\\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m$.\n",
      "\n",
      "1.0\n",
      "For each integer $n\\geq3$, let $f(n)$ be the number of $3$-element subsets of the vertices of the regular $n$-gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of $n$ such that $f(n+1)=f(n)+78$.\n",
      "Found similar problem: For each integer $n\\geq3$, let $f(n)$ be the number of $3$-element subsets of the vertices of the regular $n$-gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of $n$ such that $f(n+1)=f(n)+78$.\n",
      "\n",
      "1.0000001192092896\n",
      "Find the least positive integer $n$ such that when $3^n$ is written in base $143$, its two right-most digits in base $143$ are $01$.\n",
      "Found similar problem: Find the least positive integer $n$ such that when $3^n$ is written in base $143$, its two right-most digits in base $143$ are $01$.\n",
      "\n",
      "1.0000001192092896\n",
      "Two real numbers $a$ and $b$ are chosen independently and uniformly at random from the interval $(0, 75)$. Let $O$ and $P$ be two points on the plane with $OP = 200$. Let $Q$ and $R$ be on the same side of line $OP$ such that the degree measures of $\\angle POQ$ and $\\angle POR$ are $a$ and $b$ respectively, and $\\angle OQP$ and $\\angle ORP$ are both right angles. The probability that $QR \\leq 100$ is equal to $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.\n",
      "Found similar problem: Two real numbers $a$ and $b$ are chosen independently and uniformly at random from the interval $(0, 75)$. Let $O$ and $P$ be two points on the plane with $OP = 200$. Let $Q$ and $R$ be on the same side of line $OP$ such that the degree measures of $\\angle POQ$ and $\\angle POR$ are $a$ and $b$ respectively, and $\\angle OQP$ and $\\angle ORP$ are both right angles. The probability that $QR \\leq 100$ is equal to $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.\n",
      "\n",
      "1.0\n",
      "Consider arrangements of the $9$ numbers $1, 2, 3, \\dots, 9$ in a $3 \\times 3$ array. For each such arrangement, let $a_1$, $a_2$, and $a_3$ be the medians of the numbers in rows $1$, $2$, and $3$ respectively, and let $m$ be the median of $\\{a_1, a_2, a_3\\}$. Let $Q$ be the number of arrangements for which $m = 5$. Find the remainder when $Q$ is divided by $1000$.\n",
      "Found similar problem: Consider arrangements of the $9$ numbers $1, 2, 3, \\dots, 9$ in a $3 \\times 3$ array. For each such arrangement, let $a_1$, $a_2$, and $a_3$ be the medians of the numbers in rows $1$, $2$, and $3$ respectively, and let $m$ be the median of $\\{a_1, a_2, a_3\\}$. Let $Q$ be the number of arrangements for which $m = 5$. Find the remainder when $Q$ is divided by $1000$.\n",
      "\n",
      "0.8095523118972778\n",
      "A small [ square](https://artofproblemsolving.com/wiki/index.php/Square_(geometry)) is constructed inside a square of [area](https://artofproblemsolving.com/wiki/index.php/Area) 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the [ vertices](https://artofproblemsolving.com/wiki/index.php/Vertex) to the division points closest to the opposite vertices. Find the value of $n$ if the the [area](https://artofproblemsolving.com/wiki/index.php/Area) of the small square is exactly $\\frac1{1985}$. \n",
      "[AIME 1985 Problem 4.png](https://artofproblemsolving.com/wiki/index.php/File:AIME_1985_Problem_4.png)\n",
      "Found similar problem: A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of $n$ if the the area of the small square is exactly $\\frac1{1985}$.\n",
      "\n",
      "1.0\n",
      "For each ordered pair of real numbers $(x,y)$ satisfying \\[\\log_2(2x+y) = \\log_4(x^2+xy+7y^2)\\]there is a real number $K$ such that \\[\\log_3(3x+y) = \\log_9(3x^2+4xy+Ky^2).\\]Find the product of all possible values of $K$.\n",
      "Found similar problem: For each ordered pair of real numbers $(x,y)$ satisfying \\[\\log_2(2x+y) = \\log_4(x^2+xy+7y^2)\\]there is a real number $K$ such that \\[\\log_3(3x+y) = \\log_9(3x^2+4xy+Ky^2).\\]Find the product of all possible values of $K$.\n",
      "\n",
      "1.0\n",
      "The solutions to the system of equations\n",
      "\n",
      "$\\log_{225}x+\\log_{64}y=4$\n",
      "$\\log_{x}225-\\log_{y}64=1$\n",
      "are $(x_1,y_1)$ and $(x_2,y_2)$. Find $\\log_{30}\\left(x_1y_1x_2y_2\\right)$.\n",
      "Found similar problem: The solutions to the system of equations\n",
      "$\\log_{225}x+\\log_{64}y=4$\n",
      "$\\log_{x}225-\\log_{y}64=1$\n",
      " are $(x_1,y_1)$ and $(x_2,y_2)$. Find $\\log_{30}\\left(x_1y_1x_2y_2\\right)$.\n",
      "\n",
      "0.9157161116600037\n",
      "There is a prime number $p$ such that $16p+1$ is the cube of a positive integer.  Find $p$.\n",
      "Found similar problem: There is a prime number $p$ such that $16p+1$ is the cube of a positive integer. Find $p$.\n",
      " ~ pi_is_3.14\n",
      "\n",
      "0.9567503929138184\n",
      "In the expansion of $(ax + b)^{2000},$ where $a$ and $b$ are [relatively prime](https://artofproblemsolving.com/wiki/index.php/Relatively_prime) positive integers, the [coefficients](https://artofproblemsolving.com/wiki/index.php/Coefficient) of $x^{2}$ and $x^{3}$ are equal. Find $a + b$.\n",
      "Found similar problem: In the expansion of $(ax + b)^{2000},$ where $a$ and $b$ are relatively prime positive integers, the coefficients of $x^{2}$ and $x^{3}$ are equal. Find $a + b$.\n",
      "\n",
      "1.0000001192092896\n",
      "Segments $\\overline{AB}, \\overline{AC},$ and $\\overline{AD}$ are edges of a cube and $\\overline{AG}$ is a diagonal through the center of the cube. Point $P$ satisfies $BP=60\\sqrt{10}$, $CP=60\\sqrt{5}$, $DP=120\\sqrt{2}$, and $GP=36\\sqrt{7}$. Find $AP.$\n",
      "Found similar problem: Segments $\\overline{AB}, \\overline{AC},$ and $\\overline{AD}$ are edges of a cube and $\\overline{AG}$ is a diagonal through the center of the cube. Point $P$ satisfies $BP=60\\sqrt{10}$, $CP=60\\sqrt{5}$, $DP=120\\sqrt{2}$, and $GP=36\\sqrt{7}$. Find $AP.$\n",
      "\n",
      "1.0\n",
      "An $m\\times n\\times p$ rectangular box has half the volume of an $(m + 2)\\times(n + 2)\\times(p + 2)$ rectangular box, where $m, n,$ and $p$ are integers, and $m\\le n\\le p.$  What is the largest possible value of $p$?\n",
      "Found similar problem: An $m\\times n\\times p$ rectangular box has half the volume of an $(m + 2)\\times(n + 2)\\times(p + 2)$ rectangular box, where $m, n,$ and $p$ are integers, and $m\\le n\\le p.$ What is the largest possible value of $p$?\n",
      "\n",
      "1.0000001192092896\n",
      "Determine $3x_4+2x_5$ if $x_1$, $x_2$, $x_3$, $x_4$, and $x_5$ satisfy the system of equations below.\n",
      "\n",
      "$2x_1+x_2+x_3+x_4+x_5=6$\n",
      "$x_1+2x_2+x_3+x_4+x_5=12$\n",
      "$x_1+x_2+2x_3+x_4+x_5=24$\n",
      "$x_1+x_2+x_3+2x_4+x_5=48$\n",
      "$x_1+x_2+x_3+x_4+2x_5=96$\n",
      "Found similar problem: Determine $3x_4+2x_5$ if $x_1$, $x_2$, $x_3$, $x_4$, and $x_5$ satisfy the system of equations below.\n",
      "$2x_1+x_2+x_3+x_4+x_5=6$\n",
      "$x_1+2x_2+x_3+x_4+x_5=12$\n",
      "$x_1+x_2+2x_3+x_4+x_5=24$\n",
      "$x_1+x_2+x_3+2x_4+x_5=48$\n",
      "$x_1+x_2+x_3+x_4+2x_5=96$\n",
      "\n",
      "1.0000001192092896\n",
      "The probability that a set of three distinct vertices chosen at random from among the vertices of a regular n-gon determine an obtuse triangle is $\\frac{93}{125}$ . Find the sum of all possible values of $n$.\n",
      "Found similar problem: The probability that a set of three distinct vertices chosen at random from among the vertices of a regular n-gon determine an obtuse triangle is $\\frac{93}{125}$ . Find the sum of all possible values of $n$.\n",
      "\n",
      "1.0\n",
      "Let $SP_1P_2P_3EP_4P_5$ be a heptagon. A frog starts jumping at vertex $S$. From any vertex of the heptagon except $E$, the frog may jump to either of the two adjacent vertices. When it reaches vertex $E$, the frog stops and stays there. Find the number of distinct sequences of jumps of no more than $12$ jumps that end at $E$.\n",
      "Found similar problem: Let $SP_1P_2P_3EP_4P_5$ be a heptagon. A frog starts jumping at vertex $S$. From any vertex of the heptagon except $E$, the frog may jump to either of the two adjacent vertices. When it reaches vertex $E$, the frog stops and stays there. Find the number of distinct sequences of jumps of no more than $12$ jumps that end at $E$.\n",
      "\n",
      "0.9327705502510071\n",
      "Find the smallest prime that is the fifth term of an increasing [arithmetic sequence](https://artofproblemsolving.com/wiki/index.php/Arithmetic_sequence), all four preceding terms also being [prime](https://artofproblemsolving.com/wiki/index.php/Prime_number).\n",
      "Found similar problem: Find the smallest prime that is the fifth term of an increasing arithmetic sequence, all four preceding terms also being prime.\n",
      "\n",
      "0.9350147247314453\n",
      "Maya lists all the positive divisors of $2010^2$. She then randomly selects two distinct divisors from this list. Let $p$ be the [probability](https://artofproblemsolving.com/wiki/index.php/Probability) that exactly one of the selected divisors is a [perfect square](https://artofproblemsolving.com/wiki/index.php/Perfect_square). The probability $p$ can be expressed in the form $\\frac {m}{n}$, where $m$ and $n$ are [relatively prime](https://artofproblemsolving.com/wiki/index.php/Relatively_prime) positive integers. Find $m + n$.\n",
      "Found similar problem: Maya lists all the positive divisors of $2010^2$. She then randomly selects two distinct divisors from this list. Let $p$ be the probability that exactly one of the selected divisors is a perfect square. The probability $p$ can be expressed in the form $\\frac {m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.\n",
      "\n",
      "1.0000001192092896\n",
      "Let $S$ be a list of positive integers--not necessarily distinct--in which the number $68$ appears. The average (arithmetic mean) of the numbers in $S$ is $56$. However, if $68$ is removed, the average of the remaining numbers drops to $55$. What is the largest number that can appear in $S$?\n",
      "Found similar problem: Let $S$ be a list of positive integers--not necessarily distinct--in which the number $68$ appears. The average (arithmetic mean) of the numbers in $S$ is $56$. However, if $68$ is removed, the average of the remaining numbers drops to $55$. What is the largest number that can appear in $S$?\n",
      "\n",
      "1.0\n",
      "A triangle has vertices $A(0,0)$, $B(12,0)$, and $C(8,10)$. The probability that a randomly chosen point inside the triangle is closer to vertex $B$ than to either vertex $A$ or vertex $C$ can be written as $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.\n",
      "Found similar problem: A triangle has vertices $A(0,0)$, $B(12,0)$, and $C(8,10)$. The probability that a randomly chosen point inside the triangle is closer to vertex $B$ than to either vertex $A$ or vertex $C$ can be written as $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.\n",
      "\n",
      "1.0\n",
      "Six men and some number of women stand in a line in random order. Let $p$ be the probability that a group of at least four men stand together in the line, given that every man stands next to at least one other man. Find the least number of women in the line such that $p$ does not exceed 1 percent.\n",
      "Found similar problem: Six men and some number of women stand in a line in random order. Let $p$ be the probability that a group of at least four men stand together in the line, given that every man stands next to at least one other man. Find the least number of women in the line such that $p$ does not exceed 1 percent.\n",
      "\n",
      "0.9999999403953552\n",
      "Let $S$ be the set of positive integers $N$ with the property that the last four digits of $N$ are $2020,$ and when the last four digits are removed, the result is a divisor of $N.$ For example, $42,020$ is in $S$ because $4$ is a divisor of $42,020.$ Find the sum of all the digits of all the numbers in $S.$ For example, the number $42,020$ contributes $4+2+0+2+0=8$ to this total.\n",
      "Found similar problem: Let $S$ be the set of positive integers $N$ with the property that the last four digits of $N$ are $2020,$ and when the last four digits are removed, the result is a divisor of $N.$ For example, $42,020$ is in $S$ because $4$ is a divisor of $42,020.$ Find the sum of all the digits of all the numbers in $S.$ For example, the number $42,020$ contributes $4+2+0+2+0=8$ to this total.\n",
      "\n",
      "0.9365818500518799\n",
      "Let $S$ be the set of [ordered pairs](https://artofproblemsolving.com/wiki/index.php/Ordered_pair) $(x, y)$ such that $0 < x \\le 1, 0<y\\le 1,$ and $\\left[\\log_2{\\left(\\frac 1x\\right)}\\right]$ and $\\left[\\log_5{\\left(\\frac 1y\\right)}\\right]$ are both even. Given that the area of the graph of $S$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$ The notation $[z]$ denotes the [greatest integer](https://artofproblemsolving.com/wiki/index.php/Floor_function) that is less than or equal to $z.$\n",
      "Found similar problem: Let $S$ be the set of ordered pairs $(x, y)$ such that $0 < x \\le 1, 0<y\\le 1,$ and $\\left[\\log_2{\\left(\\frac 1x\\right)}\\right]$ and $\\left[\\log_5{\\left(\\frac 1y\\right)}\\right]$ are both even. Given that the area of the graph of $S$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$ The notation $[z]$ denotes the greatest integer that is less than or equal to $z.$\n",
      "\n",
      "1.0\n",
      "Find the number of rational numbers $r$, $0<r<1$, such that when $r$ is written as a fraction in lowest terms, the numerator and the denominator have a sum of 1000.\n",
      "Found similar problem: Find the number of rational numbers $r$, $0<r<1$, such that when $r$ is written as a fraction in lowest terms, the numerator and the denominator have a sum of 1000.\n",
      "\n",
      "1.0\n",
      "Find the number of ways $66$ identical coins can be separated into three nonempty piles so that there are fewer coins in the first pile than in the second pile and fewer coins in the second pile than in the third pile.\n",
      "Found similar problem: Find the number of ways $66$ identical coins can be separated into three nonempty piles so that there are fewer coins in the first pile than in the second pile and fewer coins in the second pile than in the third pile.\n",
      "\n",
      "1.0\n",
      "In rectangle $ABCD$, $AB = 12$ and $BC = 10$.  Points $E$ and $F$ lie inside rectangle $ABCD$ so that $BE = 9$, $DF = 8$, $\\overline{BE} \\parallel \\overline{DF}$, $\\overline{EF} \\parallel \\overline{AB}$, and line $BE$ intersects segment $\\overline{AD}$.  The length $EF$ can be expressed in the form $m \\sqrt{n} - p$, where $m$, $n$, and $p$ are positive integers and $n$ is not divisible by the square of any prime.  Find $m + n + p$.\n",
      "Found similar problem: In rectangle $ABCD$, $AB = 12$ and $BC = 10$. Points $E$ and $F$ lie inside rectangle $ABCD$ so that $BE = 9$, $DF = 8$, $\\overline{BE} \\parallel \\overline{DF}$, $\\overline{EF} \\parallel \\overline{AB}$, and line $BE$ intersects segment $\\overline{AD}$. The length $EF$ can be expressed in the form $m \\sqrt{n} - p$, where $m$, $n$, and $p$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n + p$.\n",
      "\n",
      "0.9999999403953552\n",
      "Two distinct, real, infinite geometric series each have a sum of $1$ and have the same second term. The third term of one of the series is $1/8$, and the second term of both series can be written in the form $\\frac{\\sqrt{m}-n}p$, where $m$, $n$, and $p$ are positive integers and $m$ is not divisible by the square of any prime. Find $100m+10n+p$.\n",
      "Found similar problem: Two distinct, real, infinite geometric series each have a sum of $1$ and have the same second term. The third term of one of the series is $1/8$, and the second term of both series can be written in the form $\\frac{\\sqrt{m}-n}p$, where $m$, $n$, and $p$ are positive integers and $m$ is not divisible by the square of any prime. Find $100m+10n+p$.\n",
      "\n",
      "1.0000001192092896\n",
      "For a permutation $p = (a_1,a_2,\\ldots,a_9)$ of the digits $1,2,\\ldots,9$, let $s(p)$ denote the sum of the three $3$-digit numbers $a_1a_2a_3$, $a_4a_5a_6$, and $a_7a_8a_9$. Let $m$ be the minimum value of $s(p)$ subject to the condition that the units digit of $s(p)$ is $0$. Let $n$ denote the number of permutations $p$ with $s(p) = m$. Find $|m - n|$.\n",
      "Found similar problem: For a permutation $p = (a_1,a_2,\\ldots,a_9)$ of the digits $1,2,\\ldots,9$, let $s(p)$ denote the sum of the three $3$-digit numbers $a_1a_2a_3$, $a_4a_5a_6$, and $a_7a_8a_9$. Let $m$ be the minimum value of $s(p)$ subject to the condition that the units digit of $s(p)$ is $0$. Let $n$ denote the number of permutations $p$ with $s(p) = m$. Find $|m - n|$.\n",
      "\n",
      "1.0\n",
      "Let $S\\,$ be a set with six elements. In how many different ways can one select two not necessarily distinct subsets of $S\\,$ so that the union of the two subsets is $S\\,$?  The order of selection does not matter; for example, the pair of subsets $\\{a, c\\},\\{b, c, d, e, f\\}$ represents the same selection as the pair $\\{b, c, d, e, f\\},\\{a, c\\}.$\n",
      "Found similar problem: Let $S\\,$ be a set with six elements. In how many different ways can one select two not necessarily distinct subsets of $S\\,$ so that the union of the two subsets is $S\\,$? The order of selection does not matter; for example, the pair of subsets $\\{a, c\\},\\{b, c, d, e, f\\}$ represents the same selection as the pair $\\{b, c, d, e, f\\},\\{a, c\\}.$\n",
      "\n",
      "0.9750144481658936\n",
      "Octagon $ABCDEFGH$ with side lengths $AB = CD = EF = GH = 10$ and $BC = DE = FG = HA = 11$ is formed by removing 6-8-10 triangles from the corners of a $23$ $\\times$ $27$ rectangle with side $\\overline{AH}$ on a short side of the rectangle, as shown. Let $J$ be the midpoint of $\\overline{AH}$, and partition the octagon into 7 triangles by drawing segments $\\overline{JB}$, $\\overline{JC}$, $\\overline{JD}$, $\\overline{JE}$, $\\overline{JF}$, and $\\overline{JG}$. Find the area of the convex polygon whose vertices are the centroids of these 7 triangles.\n",
      "Found similar problem: Octagon $ABCDEFGH$ with side lengths $AB = CD = EF = GH = 10$ and $BC = DE = FG = HA = 11$ is formed by removing 6-8-10 triangles from the corners of a $23$ $\\times$ $27$ rectangle with side $\\overline{AH}$ on a short side of the rectangle, as shown. Let $J$ be the midpoint of $\\overline{AH}$, and partition the octagon into 7 triangles by drawing segments $\\overline{JB}$, $\\overline{JC}$, $\\overline{JD}$, $\\overline{JE}$, $\\overline{JF}$, and $\\overline{JG}$. Find the area of the convex polygon whose vertices are the centroids of these 7 triangles.\n",
      "[asy] unitsize(6); pair P = (0, 0), Q = (0, 23), R = (27, 23), SS = (27, 0); pair A = (0, 6), B = (8, 0), C = (19, 0), D = (27, 6), EE = (27, 17), F = (19, 23), G = (8, 23), J = (0, 23/2), H = (0, 17); draw(P--Q--R--SS--cycle); draw(J--B); draw(J--C); draw(J--D); draw(J--EE); draw(J--F); draw(J--G); draw(A--B); draw(H--G); real dark = 0.6; filldraw(A--B--P--cycle, gray(dark)); filldraw(H--G--Q--cycle, gray(dark)); filldraw(F--EE--R--cycle, gray(dark)); filldraw(D--C--SS--cycle, gray(dark)); dot(A); dot(B); dot(C); dot(D); dot(EE); dot(F); dot(G); dot(H); dot(J); dot(H); defaultpen(fontsize(10pt)); real r = 1.3; label(\"$A$\", A, W*r); label(\"$B$\", B, S*r); label(\"$C$\", C, S*r); label(\"$D$\", D, E*r); label(\"$E$\", EE, E*r); label(\"$F$\", F, N*r); label(\"$G$\", G, N*r); label(\"$H$\", H, W*r); label(\"$J$\", J, W*r); [/asy]\n",
      "\n",
      "1.0\n",
      "The 8 eyelets for the lace of a sneaker all lie on a rectangle, four equally spaced on each of the longer sides. The rectangle has a width of 50 mm and a length of 80 mm. There is one eyelet at each vertex of the rectangle. The lace itself must pass between the vertex eyelets along a width side of the rectangle and then crisscross between successive eyelets until it reaches the two eyelets at the other width side of the rectangle as shown. After passing through these final eyelets, each of the ends of the lace must extend at least 200 mm farther to allow a knot to be tied. Find the minimum length of the lace in millimeters.\n",
      "[asy] size(200); defaultpen(linewidth(0.7)); path laceL=(-20,-30)..tension 0.75 ..(-90,-135)..(-102,-147)..(-152,-150)..tension 2 ..(-155,-140)..(-135,-40)..(-50,-4)..tension 0.8 ..origin; path laceR=reflect((75,0),(75,-240))*laceL; draw(origin--(0,-240)--(150,-240)--(150,0)--cycle,gray); for(int i=0;i<=3;i=i+1) { path circ1=circle((0,-80*i),5),circ2=circle((150,-80*i),5); unfill(circ1); draw(circ1); unfill(circ2); draw(circ2); } draw(laceL--(150,-80)--(0,-160)--(150,-240)--(0,-240)--(150,-160)--(0,-80)--(150,0)^^laceR,linewidth(1));[/asy]\n",
      "Found similar problem: The 8 eyelets for the lace of a sneaker all lie on a rectangle, four equally spaced on each of the longer sides. The rectangle has a width of 50 mm and a length of 80 mm. There is one eyelet at each vertex of the rectangle. The lace itself must pass between the vertex eyelets along a width side of the rectangle and then crisscross between successive eyelets until it reaches the two eyelets at the other width side of the rectangle as shown. After passing through these final eyelets, each of the ends of the lace must extend at least 200 mm farther to allow a knot to be tied. Find the minimum length of the lace in millimeters.\n",
      "[asy] size(200); defaultpen(linewidth(0.7)); path laceL=(-20,-30)..tension 0.75 ..(-90,-135)..(-102,-147)..(-152,-150)..tension 2 ..(-155,-140)..(-135,-40)..(-50,-4)..tension 0.8 ..origin; path laceR=reflect((75,0),(75,-240))*laceL; draw(origin--(0,-240)--(150,-240)--(150,0)--cycle,gray); for(int i=0;i<=3;i=i+1) { path circ1=circle((0,-80*i),5),circ2=circle((150,-80*i),5); unfill(circ1); draw(circ1); unfill(circ2); draw(circ2); } draw(laceL--(150,-80)--(0,-160)--(150,-240)--(0,-240)--(150,-160)--(0,-80)--(150,0)^^laceR,linewidth(1));[/asy]\n",
      "\n",
      "0.97706139087677\n",
      "An integer is called snakelike if its decimal representation $a_1a_2a_3\\cdots a_k$ satisfies $a_i<a_{i+1}$ if $i$ is [ odd](https://artofproblemsolving.com/wiki/index.php/Odd_integer) and $a_i>a_{i+1}$ if $i$ is [ even](https://artofproblemsolving.com/wiki/index.php/Even_integer). How many snakelike integers between 1000 and 9999 have four distinct digits?\n",
      "Found similar problem: An integer is called snakelike if its decimal representation $a_1a_2a_3\\cdots a_k$ satisfies $a_i<a_{i+1}$ if $i$ is odd and $a_i>a_{i+1}$ if $i$ is even. How many snakelike integers between 1000 and 9999 have four distinct digits?\n",
      "\n",
      "1.0\n",
      "Consider the sequence $(a_k)_{k\\ge 1}$ of positive rational numbers defined by $a_1 = \\frac{2020}{2021}$ and for $k\\ge 1$, if $a_k = \\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then\n",
      "\\[a_{k+1} = \\frac{m + 18}{n+19}.\\]Determine the sum of all positive integers $j$ such that the rational number $a_j$ can be written in the form $\\frac{t}{t+1}$ for some positive integer $t$.\n",
      "Found similar problem: Consider the sequence $(a_k)_{k\\ge 1}$ of positive rational numbers defined by $a_1 = \\frac{2020}{2021}$ and for $k\\ge 1$, if $a_k = \\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then\n",
      "\\[a_{k+1} = \\frac{m + 18}{n+19}.\\]Determine the sum of all positive integers $j$ such that the rational number $a_j$ can be written in the form $\\frac{t}{t+1}$ for some positive integer $t$.\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# Filter for AMC only problems.\n",
    "amc_dataset = []\n",
    "for row in amc_aime:\n",
    "    problem = row['problem']\n",
    "    result_dict = rag_server.top_k(problem, k=1)[0]\n",
    "    score = result_dict['score']\n",
    "    if score > 0.9 or 'AIME' in problem or 'aime' in problem:\n",
    "        print(score)\n",
    "        print(problem)\n",
    "        print(\"Found similar problem:\", result_dict['text'])\n",
    "    else:\n",
    "        amc_dataset.append(row)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Refine AMC Problems for No Multiple Choice (Direct Answer)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "import ast\n",
    "import re\n",
    "\n",
    "from deepscaler.utils import call_gemini_llm\n",
    "from deepscaler.system_prompts import REFINE_AMC_PROMPT, FETCH_MC_PROMPT\n",
    "\n",
    "def parse_llm_output(llm_output: str) -> dict:\n",
    "    try:\n",
    "        # Remove code fences in case they appear\n",
    "        llm_output = repr(llm_output)\n",
    "        cleaned = re.sub(r'```(?:json)?', '', llm_output)\n",
    "        cleaned = re.sub(r'```', '', cleaned)\n",
    "        cleaned = cleaned.strip()\n",
    "        # Parse as Python dictionary\n",
    "        parsed_dict = ast.literal_eval(cleaned)\n",
    "        parsed_dict = ast.literal_eval(parsed_dict)\n",
    "        return parsed_dict\n",
    "    except:\n",
    "        print(\"FAIL PARSING\")\n",
    "        print(llm_output)\n",
    "        return {\n",
    "            'problem': None,\n",
    "            'A': None,\n",
    "            'B': None,\n",
    "            'C': None,\n",
    "            'D': None,\n",
    "            'E': None\n",
    "        }\n",
    "\n",
    "def process_entry_no_mc(entry):\n",
    "    output_dict = {}\n",
    "    # 1) Get the problem text\n",
    "    problem_text = entry['problem']\n",
    "    solution_text = entry['solution']\n",
    "    # 2) Call Gemini LLM\n",
    "    output_str = call_gemini_llm(problem_text, system_prompt=REFINE_AMC_PROMPT)\n",
    "    if not output_str:\n",
    "        print(\"Gemini not happy.\")\n",
    "        return {}\n",
    "    # 3) Parse the LLM output into a Python dict\n",
    "    python_dict = parse_llm_output(output_str)\n",
    "    python_dict = dict(python_dict)\n",
    "    output_dict['problem'] = python_dict['problem']\n",
    "    output_dict['solution'] = entry['solution']\n",
    "    if python_dict.get('A', None) is None and python_dict.get('B', None):\n",
    "        return {}\n",
    "    answer = call_gemini_llm(f'Problem: {problem_text} \\n Solution: {solution_text}', system_prompt=FETCH_MC_PROMPT)\n",
    "    answer = answer.upper()\n",
    "    if len(answer) > 1 or answer not in ['A', 'B', 'C', 'D', 'E']:\n",
    "        answer = call_gemini_llm(f'Problem: {problem_text} \\n Solution: {solution_text}', system_prompt=FETCH_MC_PROMPT)\n",
    "        print(\"Retrying answer fetching:\")\n",
    "        print(answer)\n",
    "        answer = answer.upper()\n",
    "    if answer not in ['A', 'B', 'C', 'D', 'E'] or answer not in python_dict:\n",
    "        print('Failed extracting answer')\n",
    "        print(problem_text)\n",
    "        return {}\n",
    "    output_dict['answer'] = python_dict[answer]\n",
    "    return output_dict"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n",
      "huggingface/tokenizers: The current process just got forked, after parallelism has already been used. Disabling parallelism to avoid deadlocks...\n",
      "To disable this warning, you can either:\n",
      "\t- Avoid using `tokenizers` before the fork if possible\n",
      "\t- Explicitly set the environment variable TOKENIZERS_PARALLELISM=(true | false)\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Which of the following is equivalent to \\\\\"If P is true, then Q is false.\\\\\"?\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Figure 1 is called a \\\\\"stack map.\\\\\" The numbers tell how many cubes are stacked in each position. Fig. 2 shows these cubes, and Fig. 3 shows the view of the stacked cubes as seen from the front. Which of the following is the front view for the stack map in Fig. 4?\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"A given convex pentagon $ABCDE$ has the property that the area of each of the five triangles $ABC$, $BCD$, $CDE$, $DEA$, and $EAB$ is unity. Show that all pentagons with the above property have the same area, and calculate that area. Show, furthermore, that there are infinitely many non-congruent pentagons having the above area property.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Suppose $q_1, q_2, ...$ is an infinite sequence of integers satisfying the following two conditions:\\\\n(a) $m - n$ divides $q_m - q_n$ for $m > n \\\\\\\\geq 0$\\\\n(b) There is a polynomial $P$ such that $|q_n| < P(n)$ for all $n$.\\\\nProve that there is a polynomial $Q$ such that $q_n = Q(n)$ for each $n$.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"The geometric mean of any set of $m$ non-negative numbers is the $m$-th root of their product.\\\\n$\\\\\\\\quad (\\\\\\\\text{i})\\\\\\\\quad$ For which positive integers $n$ is there a finite set $S_n$ of $n$ distinct positive integers such that the geometric mean of any subset of $S_n$ is an integer?\\\\n$\\\\\\\\quad (\\\\\\\\text{ii})\\\\\\\\quad$ Is there an infinite set $S$ of distinct positive integers such that the geometric mean of any finite subset of $S$ is an integer?\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $\\\\\\\\, a =(m^{m+1} + n^{n+1})/(m^m + n^n), \\\\\\\\,$ where $\\\\\\\\,m\\\\\\\\,$ and $\\\\\\\\,n\\\\\\\\,$ are positive integers.  Prove that $\\\\\\\\,a^m + a^n \\\\\\\\geq m^m + n^n$.\\\\n[You may wish to analyze the ratio $\\\\\\\\,(a^N - N^N)/(a-N),$ for real $\\\\\\\\, a \\\\\\\\geq 0 \\\\\\\\,$ and integer $\\\\\\\\, N \\\\\\\\geq 1$.]\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "A\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $\\, a =(m^{m+1} + n^{n+1})/(m^m + n^n), \\,$ where $\\,m\\,$ and $\\,n\\,$ are positive integers.  Prove that $\\,a^m + a^n \\geq m^m + n^n$.\n",
      "[You may wish to analyze the ratio $\\,(a^N - N^N)/(a-N),$ for real $\\, a \\geq 0 \\,$ and integer $\\, N \\geq 1$.]\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"In right triangle $ABC$ with right angle $C$, $CA = 30$ and $CB = 16$. Its legs $CA$ and $CB$ are extended beyond $A$ and $B$. Points $O_1$ and $O_2$ lie in the exterior of the triangle and are the centers of two circles with equal radii. The circle with center $O_1$ is tangent to the hypotenuse and to the extension of leg $CA$, the circle with center $O_2$ is tangent to the hypotenuse and to the extension of leg $CB$, and the circles are externally tangent to each other. The length of the radius either circle can be expressed as $p/q$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"(Titu Andreescu)\\\\nProve that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Are there integers $a$ and $b$ such that $a^5b+3$ and $ab^5+3$ are both perfect cubes of integers?\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $ABC$ be a triangle with $\\\\\\\\angle A = 90^{\\\\\\\\circ}$. Points $D$ and $E$ lie on sides $AC$ and $AB$, respectively, such that $\\\\\\\\angle ABD = \\\\\\\\angle DBC$ and $\\\\\\\\angle ACE = \\\\\\\\angle ECB$. Segments $BD$ and $CE$ meet at $I$.  Determine whether or not it is possible for segments $AB, AC, BI, ID, CI, IE$ to all have integer lengths.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Ray\\'s car averages $40$ miles per gallon of gasoline, and Tom\\'s car averages $10$ miles per gallon of gasoline. Ray and Tom each drive the same number of miles. What is the cars\\' combined rate of miles per gallon of gasoline?\",\\n\"A\": \"10\",\\n\"B\": \"16\",\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Two given circles intersect in two points $P$ and $Q$. Show how to construct a segment $AB$ passing through $P$ and terminating on the two circles such that $AP \\\\\\\\cdot PB$ is a maximum.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Let $k$ be a positive integer. Two players $A$ and $B$ play a game on an infinite grid of regular hexagons. Initially all the grid cells are empty. Then the players alternately take turns with $A$ moving first. In his move, $A$ may choose two adjacent hexagons in the grid which are empty and place a counter in both of them. In his move, $B$ may choose any counter on the board and remove it. If at any time there are $k$ consecutive grid cells in a line all of which contain a counter, $A$ wins. Find the minimum value of $k$ for which $A$ cannot win in a finite number of moves, or prove that no such minimum value exists.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```\\n'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $k={2008}^{2}+{2}^{2008}$. What is the units digit of $k^2+2^k$?\n",
      "$\\mathrm{(A)}\\ 0\\qquad\\mathrm{(B)}\\ 2\\qquad\\mathrm{(C)}\\ 4\\qquad\\mathrm{(D)}\\ 6\\qquad\\mathrm{(E)}\\ 8$\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "In rectangle $ABCD,$ $AB=6$ and $BC=3$. Point $E$ between $B$ and $C$, and point $F$ between $E$ and $C$ are such that $BE=EF=FC$. Segments $\\overline{AE}$ and $\\overline{AF}$ intersect $\\overline{BD}$ at $P$ and $Q$, respectively. The ratio $BP:PQ:QD$ can be written as $r:s:t$ where the greatest common factor of $r,s,$ and $t$ is $1.$ What is $r+s+t$?\n",
      "$\\textbf{(A) } 7 \\qquad \\textbf{(B) } 9 \\qquad \\textbf{(C) } 12 \\qquad \\textbf{(D) } 15 \\qquad \\textbf{(E) } 20$\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Two given circles intersect in two points $P$ and $Q$. Show how to construct a segment $AB$ passing through $P$ and terminating on the two circles such that $AP\\cdot PB$ is a maximum.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"(Titu Andreescu) Prove that for every nonnegative integer $n$, the number $7^{7^n}+1$ is the product of at least $2n+3$ (not necessarily distinct) primes.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $b\\\\\\\\geq 2$ be an integer, and let $s_b(n)$ denote the sum of the digits of $n$ when it is written in base $b$.  Show that there are infinitely many positive integers that cannot be represented in the form $n+s_b(n)$, where $n$ is a positive integer.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Let $a_n$ be the number of permutations $(x_1, x_2, \\\\\\\\dots, x_n)$ of the numbers $(1,2,\\\\\\\\dots, n)$ such that the $n$ ratios $\\\\\\\\frac{x_k}{k}$ for $1\\\\\\\\le k\\\\\\\\le n$ are all distinct. Prove that $a_n$ is odd for all $n\\\\\\\\ge 1.$\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $b\\geq 2$ be an integer, and let $s_b(n)$ denote the sum of the digits of $n$ when it is written in base $b$.  Show that there are infinitely many positive integers that cannot be represented in the form $n+s_b(n)$, where $n$ is a positive integer.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"A game of solitaire is played with $R$ red cards, $W$ white cards, and $B$ blue cards. A player plays all the cards one at a time. With each play he accumulates a penalty. If he plays a blue card, then he is charged a penalty which is the number of white cards still in his hand. If he plays a white card, then he is charged a penalty which is twice the number of red cards still in his hand. If he plays a red card, then he is charged a penalty which is three times the number of blue cards still in his hand. Find, as a function of $R, W,$ and $B,$ the minimal total penalty a player can amass and all the ways in which this minimum can be achieved.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Consider the assertion that for each positive integer $n \\\\\\\\ge 2$, the remainder upon dividing $2^{2^n}$ by $2^n-1$ is a power of 4. Either prove the assertion or find (with proof) a counter-example.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $A_1, A_2, ..., A_{n+1}$ be distinct subsets of $[n]$ with $|A_1| = |A_2| = \\\\\\\\cdots = |A_{n+1}| = 3$. Prove that $|A_i \\\\\\\\cap A_j| = 1$ for some pair $\\\\\\\\{i, j\\\\\\\\}$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"As indicated by the diagram below, a rectangular piece of paper is folded bottom to top, then left to right, and finally, a hole is punched at X.  What does the paper look like when unfolded?\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $a_n$ be the number of permutations $(x_1, x_2, \\dots, x_n)$ of the numbers $(1,2,\\dots, n)$ such that the $n$ ratios $\\frac{x_k}{k}$ for $1\\le k\\le n$ are all distinct. Prove that $a_n$ is odd for all $n\\ge 1.$\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Which pattern of identical squares could NOT be folded along the lines shown to form a cube?\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $A_1,A_2,...,A_{n+1}$ be distinct subsets of $[n]$ with $|A_1|=|A_2|=\\cdots =|A_{n+1}|=3$.  Prove that $|A_i\\cap A_j|=1$ for some pair $\\{i,j\\}$.\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "In quadrilateral $ABCD$, $m\\angle B = m \\angle C = 120^{\\circ}, AB=3, BC=4,$ and $CD=5.$ Find the area of $ABCD.$\n",
      "$\\text{(A) }15 \\qquad \\text{(B) }9 \\sqrt{3} \\qquad \\text{(C) }\\frac{45 \\sqrt{3}}{4} \\qquad \\text{(D) }\\frac{47 \\sqrt{3}}{4} \\qquad \\text{(E) }15 \\sqrt{3}$\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"([Sam Vandervelde](https://artofproblemsolving.com/wiki/index.php/Sam_Vandervelde)) At a certain mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an even number of his or her friends. Prove that the number of ways that the mathematicians may be split between the two rooms is a power of two (i.e., is of the form $2^k$ for some positive integer $k$).\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Two congruent 30-60-90 are placed so that they overlap partly and their hypotenuses coincide. If the hypotenuse of each triangle is 12, the area common to both triangles is \n",
      " $\\textbf{(A)}\\ 6\\sqrt3\\qquad\\textbf{(B)}\\ 8\\sqrt3\\qquad\\textbf{(C)}\\ 9\\sqrt3\\qquad\\textbf{(D)}\\ 12\\sqrt3\\qquad\\textbf{(E)}\\ 24$\n",
      "Retrying answer fetching:\n",
      "B\n",
      "Retrying answer fetching:\n",
      "B\n",
      "Retrying answer fetching:\n",
      "A\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Prove that for each $n \\\\\\\\geq 2$, there is a set $S$ of $n$ integers such that $(a-b)^2$ divides $ab$ for every distinct $a,b \\\\\\\\in S$.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"The isosceles triangle $\\\\\\\\triangle ABC$, with $AB=AC$, is inscribed in the circle $\\\\\\\\omega$. Let $P$ be a variable point on the arc $\\\\\\\\stackrel{\\\\\\\\frown}{BC}$ that does not contain $A$, and let $I_B$ and $I_C$ denote the incenters of triangles $\\\\\\\\triangle ABP$ and $\\\\\\\\triangle ACP$, respectively.\\\\nProve that as $P$ varies, the circumcircle of triangle $\\\\\\\\triangle PI_BI_C$ passes through a fixed point.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Prove that for each $n\\geq 2$, there is a set $S$ of $n$ integers such that $(a-b)^2$ divides $ab$ for every distinct $a,b\\in S$.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $A_1A_2A_3$ be a triangle and let $\\\\\\\\omega_1$ be a circle in its plane passing through $A_1$ and $A_2.$ Suppose there exist circles $\\\\\\\\omega_2, \\\\\\\\omega_3, \\\\\\\\dots, \\\\\\\\omega_7$ such that for $k = 2, 3, \\\\\\\\dots, 7,$ $\\\\\\\\omega_k$ is externally tangent to $\\\\\\\\omega_{k - 1}$ and passes through $A_k$ and $A_{k + 1},$  where $A_{n + 3} = A_{n}$ for all $n \\\\\\\\ge 1$. Prove that $\\\\\\\\omega_7 = \\\\\\\\omega_1.$\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $s_1, s_2, s_3, \\\\\\\\ldots$ be an infinite, nonconstant sequence of rational numbers, meaning it is not the case that $s_1 = s_2 = s_3 = \\\\\\\\cdots.$  Suppose that $t_1, t_2, t_3, \\\\\\\\ldots$ is also an infinite, nonconstant sequence of rational numbers with the property that $(s_i - s_j)(t_i - t_j)$ is an integer for all $i$ and $j$.  Prove that there exists a rational number $r$ such that $(s_i - s_j)r$ and $(t_i - t_j)/r$ are integers for all $i$ and $j$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $p$ be an odd prime. The sequence $(a_n)_{n \\\\\\\\geq 0}$ is defined as follows: $a_0 = 0$, $a_1 = 1$, $\\\\\\\\ldots$, $a_{p-2} = p-2$ and, for all $n \\\\\\\\geq p-1$, $a_n$ is the least positive integer that does not form an arithmetic sequence of length $p$ with any of the preceding terms. Prove that, for all $n$, $a_n$ is the number obtained by writing $n$ in base $p-1$ and reading the result in base $p$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $s_1, s_2, s_3, \\ldots$ be an infinite, nonconstant sequence of rational numbers, meaning it is not the case that $s_1 = s_2 = s_3 = \\cdots.$  Suppose that $t_1, t_2, t_3, \\ldots$ is also an infinite, nonconstant sequence of rational numbers with the property that $(s_i - s_j)(t_i - t_j)$ is an integer for all $i$ and $j$.  Prove that there exists a rational number $r$ such that $(s_i - s_j)r$ and $(t_i - t_j)/r$ are integers for all $i$ and $j$.\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Jo and Blair take turns counting from $1$ to one more than the last number said by the other person. Jo starts by saying , so Blair follows by saying . Jo then says , and so on. What is the $53^{\\text{rd}}$ number said?\n",
      "$\\textbf{(A)}\\ 2 \\qquad \\textbf{(B)}\\ 3 \\qquad \\textbf{(C)}\\ 5 \\qquad \\textbf{(D)}\\ 6 \\qquad \\textbf{(E)}\\ 8$\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"(Kiran Kedlaya) Let $p$ be a prime number and let $s$ be an integer with $0 < s < p$. Prove that there exist integers $m$ and $n$ with $0 < m < n < p$ and\\\\n\\\\n$\\\\\\\\left\\\\\\\\{ \\\\\\\\frac{sm}{p} \\\\\\\\right\\\\\\\\} < \\\\\\\\left\\\\\\\\{ \\\\\\\\frac{sn}{p} \\\\\\\\right\\\\\\\\} < \\\\\\\\frac{s}{p}$\\\\n\\\\nif and only if $s$ is not a divisor of $p-1$.\\\\n\\\\nNote: For $x$ a real number, let $\\\\\\\\lfloor x \\\\\\\\rfloor$ denote the greatest integer less than or equal to $x$, and let $\\\\\\\\{x\\\\\\\\} = x - \\\\\\\\lfloor x \\\\\\\\rfloor$ denote the fractional part of $x$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Gemini not happy.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Tess runs counterclockwise around rectangular block $JKLM$. She lives at corner $J$. Which graph could represent her straight-line distance from home?\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "(Kiran Kedlaya) Let $p$ be a prime number and let $s$ be an integer with $0 < s < p$. Prove that there exist integers $m$ and $n$ with $0 < m < n < p$ and\n",
      "\n",
      "\n",
      "$\\left\\{ \\frac{sm}{p} \\right\\} < \\left\\{ \\frac{sn}{p} \\right\\} < \\frac{s}{p}$\n",
      "\n",
      "\n",
      "if and only if $s$ is not a divisor of $p-1$.\n",
      "Note: For $x$ a real number, let $\\lfloor x \\rfloor$ denote the greatest integer less than or equal to $x$, and let $\\{x\\} = x - \\lfloor x \\rfloor$ denote the fractional part of $x$.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $\\\\\\\\, a_1, a_2, a_3, \\\\\\\\ldots \\\\\\\\,$ be a sequence of positive real numbers satisfying $\\\\\\\\, \\\\\\\\sum_{j = 1}^n a_j \\\\\\\\geq \\\\\\\\sqrt {n} \\\\\\\\,$ for all $\\\\\\\\, n \\\\\\\\geq 1$. Prove that, for all $\\\\\\\\, n \\\\\\\\geq 1, \\\\\\\\,$ \\\\n\\\\\\\\[\\\\\\\\sum_{j = 1}^n a_j^2 > \\\\\\\\frac {1}{4} \\\\\\\\left( 1 + \\\\\\\\frac {1}{2} + \\\\\\\\cdots + \\\\\\\\frac {1}{n} \\\\\\\\right).\\\\\\\\]\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Gemini not happy.\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $\\, a_1, a_2, a_3, \\ldots \\,$ be a sequence of positive real numbers satisfying $\\, \\sum_{j = 1}^n a_j \\geq \\sqrt {n} \\,$ for all $\\, n \\geq 1$. Prove that, for all $\\, n \\geq 1, \\,$\n",
      "\\[\\sum_{j = 1}^n a_j^2 > \\frac {1}{4} \\left( 1 + \\frac {1}{2} + \\cdots + \\frac {1}{n} \\right).\\]\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Let $ABCD$ be a convex quadrilateral inscribed in a circle and satisfying $DA < AB = BC < CD$. Points $E$ and $F$ are chosen on sides $CD$ and $AB$ such that $BE \\\\\\\\perp AC$ and $EF \\\\\\\\parallel BC$. Prove that $FB = FD$.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $k={2008}^{2}+{2}^{2008}$. What is the units digit of $k^2+2^k$?\n",
      "$\\mathrm{(A)}\\ 0\\qquad\\mathrm{(B)}\\ 2\\qquad\\mathrm{(C)}\\ 4\\qquad\\mathrm{(D)}\\ 6\\qquad\\mathrm{(E)}\\ 8$\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $ABCD$ be a convex quadrilateral inscribed in a circle and satisfying $DA < AB = BC < CD$. Points $E$ and $F$ are chosen on sides $CD$ and $AB$ such that $BE \\perp AC$ and $EF \\parallel BC$. Prove that $FB = FD$.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"ABCD and A\\'B\\'C\\'D\\' are square maps of the same region, drawn to different scales and superimposed as shown in the figure. Prove that there is only one point O on the small map that lies directly over point O\\' of the large map such that O and O\\' each represent the same place of the country. Also, give a Euclidean construction (straight edge and compass) for O.\\\\n[asy] defaultpen(linewidth(0.7)+fontsize(10)); real theta = -100, r = 0.3; pair D2 = (0.3,0.76); string[] lbl = {\\'A\\', \\'B\\', \\'C\\', \\'D\\'}; draw(unitsquare); draw(shift(D2)*rotate(theta)*scale(r)*unitsquare); for(int i = 0; i < lbl.length; ++i) { pair Q = dir(135-90*i), P = (.5,.5)+Q/2^.5; label(\"$\"+lbl[i]+\"\\'$\", P, Q); label(\"$\"+lbl[i]+\"$\",D2+rotate(theta)*(r*P), rotate(theta)*Q); }[/asy]\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "$ABCD$ and $A'B'C'D'$ are square maps of the same region, drawn to different scales and superimposed as shown in the figure. Prove that there is only one point $O$ on the small map that lies directly over point $O'$ of the large map such that $O$ and $O'$ each represent the same place of the country. Also, give a Euclidean construction (straight edge and compass) for $O$.\n",
      "[asy] defaultpen(linewidth(0.7)+fontsize(10)); real theta = -100, r = 0.3; pair D2 = (0.3,0.76); string[] lbl = {'A', 'B', 'C', 'D'}; draw(unitsquare); draw(shift(D2)*rotate(theta)*scale(r)*unitsquare); for(int i = 0; i < lbl.length; ++i) { pair Q = dir(135-90*i), P = (.5,.5)+Q/2^.5; label(\"$\"+lbl[i]+\"'$\", P, Q); label(\"$\"+lbl[i]+\"$\",D2+rotate(theta)*(r*P), rotate(theta)*Q); }[/asy]\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter $AB$. Denote by $P, Q, R, S$ the feet of the perpendiculars from $Y$ onto lines $AX, BX, AZ, BZ$, respectively. Prove that the acute angle formed by lines $PQ$ and $RS$ is half the size of $\\\\\\\\angle XOZ$, where $O$ is the midpoint of segment $AB$.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICEFAIL PARSING\n",
      "Failed extracting answer\n",
      "\n",
      "Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter\n",
      "$AB$. Denote by $P, Q, R, S$ the feet of the perpendiculars from $Y$ onto\n",
      "lines $AX, BX, AZ, BZ$, respectively. Prove that the acute angle\n",
      "formed by lines $PQ$ and $RS$ is half the size of $\\angle XOZ$, where\n",
      "$O$ is the midpoint of segment $AB$.'```json\\n{\\n \"problem\": \"(Gregory Galparin) Let $\\\\\\\\mathcal{P}$ be a convex polygon with $n$ sides, $n\\\\\\\\ge3$. Any set of $n - 3$ diagonals of $\\\\\\\\mathcal{P}$ that do not intersect in the interior of the polygon determine a triangulation of $\\\\\\\\mathcal{P}$ into $n - 2$ triangles. If $\\\\\\\\mathcal{P}$ is regular and there is a triangulation of $\\\\\\\\mathcal{P}$ consisting of only isosceles triangles, find all the possible values of $n$.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"$N$ is the north pole. $A$ and $B$ are points on a great circle through $N$ equidistant from $N$. $C$ is a point on the equator. Show that the great circle through $C$ and $N$ bisects the angle $ACB$ in the spherical triangle $ABC$ (a spherical triangle has great circle arcs as sides).\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"A mathematical frog jumps along the number line. The frog starts at 1, and jumps according to the following rule: if the frog is at integer $n$, then it can jump either to $n+1$ or to $n+2^{m_n+1}$ where $2^{m_n}$ is the largest power of 2 that is a factor of $n$. Show that if $k\\\\\\\\ge 2$ is a positive integer and $i$ is a nonnegative integer, then the minimum number of jumps needed to reach $2^i k$ is greater than the minimum number of jumps needed to reach $2^i$.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"An $n$-term sequence $(x_1, x_2, \\\\\\\\ldots, x_n)$ in which each term is either 0 or 1 is called a binary sequence of length $n$. Let $a_n$ be the number of binary sequences of length $n$ containing no three consecutive terms equal to 0, 1, 0 in that order. Let $b_n$ be the number of binary sequences of length $n$ that contain no four consecutive terms equal to 0, 0, 1, 1 or 1, 1, 0, 0 in that order. Prove that $b_{n+1} = 2a_n$ for all positive integers $n$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $X_1, X_2, \\\\\\\\ldots, X_{100}$ be a sequence of mutually distinct nonempty subsets of a set $S$. Any two sets $X_i$ and $X_{i+1}$ are disjoint and their union is not the whole set $S$, that is, $X_i\\\\\\\\cap X_{i+1}=\\\\\\\\emptyset$ and $X_i\\\\\\\\cup X_{i+1}\\\\\\\\neq S$, for all $i\\\\\\\\in\\\\\\\\{1, \\\\\\\\ldots, 99\\\\\\\\}$. Find the smallest possible number of elements in $S$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "$N$ is the north pole. $A$ and $B$ are points on a great circle through $N$ equidistant from $N$. $C$ is a point on the equator. Show that the great circle through $C$ and $N$ bisects the angle $ACB$ in the spherical triangle $ABC$ (a spherical triangle has great circle arcs as sides).\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "(Zoran Sunik) A mathematical frog jumps along the number line. The frog starts at 1, and jumps according to the following rule: if the frog is at integer $n$, then it can jump either to $n+1$ or to $n+2^{m_n+1}$ where $2^{m_n}$ is the largest power of 2 that is a factor of $n$. Show that if $k\\ge 2$ is a positive integer and $i$ is a nonnegative integer, then the minimum number of jumps needed to reach $2^i k$ is greater than the minimum number of jumps needed to reach $2^i$.\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "An $n$-term sequence $(x_1, x_2, \\ldots, x_n)$ in which each term is either 0 or 1 is called a binary sequence of length $n$. Let $a_n$ be the number of binary sequences of length n containing no three consecutive terms equal to 0, 1, 0 in that order. Let $b_n$ be the number of binary sequences of length $n$ that contain no four consecutive terms equal to 0, 0, 1, 1 or 1, 1, 0, 0 in that order. Prove that $b_{n+1} = 2a_n$ for all positive integers $n$.\n",
      "(proposed by Kiran Kedlaya)\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"(Melanie Wood)\\\\nAlice and Bob play a game on a 6 by 6 grid.  On his or her turn, a player chooses a rational number not yet appearing on the grid and writes it in an empty square of the grid.  Alice goes first and then the players alternate.  When all squares have numbers written in them, in each row, the square with the greatest number is colored black.  Alice wins if she can then draw a line from the top of the grid to the bottom of the grid that stays in black squares, and Bob wins if she can\\'t.  (If two squares share a vertex, Alice can draw a line from on to the other that stays in those two squares.)  Find, with proof, a winning strategy for one of the players.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Right triangles $T_1$ and $T_2$, have areas of 1 and 2, respectively. A side of $T_1$ is congruent to a side of $T_2$, and a different side of $T_1$ is congruent to a different side of $T_2$. What is the square of the product of the lengths of the other (third) sides of $T_1$ and $T_2$?\n",
      "$\\textbf{(A) }\\frac{28}{3}\\qquad\\textbf{(B) }10\\qquad\\textbf{(C) }\\frac{32}{3}\\qquad\\textbf{(D) }\\frac{34}{3}\\qquad\\textbf{(E) }12$\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"An integer is assigned to each vertex of a regular pentagon so that the sum of the five integers is 2011. A turn of a solitaire game consists of subtracting an integer $m$ from each of the integers at two neighboring vertices and adding $2m$ to the opposite vertex, which is not adjacent to either of the first two vertices. (The amount $m$ and the vertices chosen can vary from turn to turn.) The game is won at a certain vertex if, after some number of turns, that vertex has the number 2011 and the other four vertices have the number 0. Prove that for any choice of the initial integers, there is exactly one vertex at which the game can be won.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"(Zuming Feng) Let $ABC$ be an acute-angled triangle, and let $P$ and $Q$ be two points on side $BC$. Construct point $C_1$ in such a way that convex quadrilateral $APBC_1$ is cyclic, $QC_1 \\\\\\\\parallel CA$, and $C_1$ and $Q$ lie on opposite sides of line $AB$. Construct point $B_1$ in such a way that convex quadrilateral $APCB_1$ is cyclic, $QB_1 \\\\\\\\parallel BA$, and $B_1$ and $Q$  lie on opposite sides of line $AC$.  Prove that points $B_1, C_1,P$, and $Q$ lie on a circle.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $k$ be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with $k+1$ digits. Every time Bernardo writes a number, Silvia erases the last $k$ digits of it. Bernardo then writes the next perfect square, Silvia erases the last $k$ digits of it, and this process continues until the last two numbers that remain on the board differ by at least 2. Let $f(k)$ be the smallest positive integer not written on the board. For example, if $k = 1$, then the numbers that Bernardo writes are $16, 25, 36, 49, 64$, and the numbers showing on the board after Silvia erases are $1, 2, 3, 4,$ and $6$, and thus $f(1) = 5$. What is the sum of the digits of $f(2) + f(4)+ f(6) + \\dots + f(2016)$?\n",
      "$\\textbf{(A)}\\ 7986\\qquad\\textbf{(B)}\\ 8002\\qquad\\textbf{(C)}\\ 8030\\qquad\\textbf{(D)}\\ 8048\\qquad\\textbf{(E)}\\ 8064$\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "(Zuming Feng) Let $ABC$ be an acute-angled triangle, and let $P$ and $Q$ be two points on side $BC$. Construct point $C_1$ in such a way that convex quadrilateral $APBC_1$ is cyclic, $QC_1 \\parallel CA$, and $C_1$ and $Q$ lie on opposite sides of line $AB$. Construct point $B_1$ in such a way that convex quadrilateral $APCB_1$ is cyclic, $QB_1 \\parallel BA$, and $B_1$ and $Q$  lie on opposite sides of line $AC$.  Prove that points $B_1, C_1,P$, and $Q$ lie on a circle.\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "In a general triangle $ADE$ (as shown) lines $\\overline{EB}$ and $\\overline{EC}$ are drawn. Which of the following angle relations is true?\n",
      "\n",
      "$\\textbf{(A)}\\ x \\plus{} z \\equal{} a \\plus{} b\\qquad \\textbf{(B)}\\ y \\plus{} z \\equal{} a \\plus{} b\\qquad \\textbf{(C)}\\ m \\plus{} x \\equal{} w \\plus{} n\\qquad \\\\\n",
      "\\textbf{(D)}\\ x \\plus{} z \\plus{} n \\equal{} w \\plus{} c \\plus{} m\\qquad \\textbf{(E)}\\ x \\plus{} y \\plus{} n \\equal{} a \\plus{} b \\plus{} m$ (Error compiling LaTeX. Unknown error_msg)\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "A rectangle with side lengths $1{ }$ and $3,$ a square with side length $1,$ and a rectangle $R$ are inscribed inside a larger square as shown. The sum of all possible values for the area of $R$ can be written in the form $\\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. What is $m+n?$\n",
      "\n",
      "$(\\textbf{A})\\: 14\\qquad(\\textbf{B}) \\: 23\\qquad(\\textbf{C}) \\: 46\\qquad(\\textbf{D}) \\: 59\\qquad(\\textbf{E}) \\: 67$\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Two points $P$ and $Q$ lie in the interior of a regular tetrahedron $ABCD$. Prove that angle $PAQ < 60^\\\\\\\\circ$.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"What is the largest number of towns that can meet the following criteria. Each pair is directly linked by just one of air, bus or train. At least one pair is linked by air, at least one pair by bus and at least one pair by train. No town has an air link, a bus link and a train link. No three towns, $A, B, C$ are such that the links between $AB, AC$ and $BC$ are all air, all bus or all train.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Two points $P$ and $Q$ lie in the interior of a regular tetrahedron $ABCD$. Prove that angle $PAQ < 60^\\circ$.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"(Ricky Liu) For what values of \\\\\\\\(k > 0\\\\\\\\) is it possible to dissect a \\\\\\\\(1 \\\\\\\\times k\\\\\\\\) rectangle into two similar, but incongruent, polygons?\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"For a positive integer $n\\\\\\\\geq 3$ plot $n$ equally spaced points around a circle. Label one of them $A$, and place a marker at $A$. One may move the marker forward in a clockwise direction to either the next point or the point after that. Hence there are a total of $2n$ distinct moves available; two from each point. Let $a_n$ count the number of ways to advance around the circle exactly twice, beginning and ending at $A$, without repeating a move. Prove that $a_{n-1}+a_n=2^n$ for all $n\\\\\\\\geq 4$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"A convex hexagon $ABCDEF$ is inscribed in a circle such that $AB=CD=EF$ and diagonals $AD,BE$, and $CF$ are concurrent. Let $P$ be the intersection of $AD$ and $CE$. Prove that $\\\\\\\\frac{CP}{PE}=(\\\\\\\\frac{AC}{CE})^2$.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"$P$ lies between the rays $OA$ and $OB$. Find $Q$ on $OA$ and $R$ on $OB$ collinear with $P$ so that $\\\\\\\\frac{1}{PQ} + \\\\\\\\frac{1}{PR}$ is as large as possible.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"If $x, y, z$ are reals such that $0 \\\\\\\\le x, y, z \\\\\\\\le 1$, show that $\\\\\\\\frac{x}{y + z + 1} + \\\\\\\\frac{y}{z + x + 1} + \\\\\\\\frac{z}{x + y +  1} \\\\\\\\le 1 - (1 - x)(1 - y)(1 - z)$\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "For a positive integer $n\\geq 3$ plot $n$ equally spaced points around a circle.  Label one of them $A$, and place a marker at $A$.  One may move the marker forward in a clockwise direction to either the next point or the point after that.  Hence there are a total of $2n$ distinct moves available; two from each point.  Let $a_n$ count the number of ways to advance around the circle exactly twice, beginning and ending at $A$, without repeating a move.  Prove that $a_{n-1}+a_n=2^n$ for all $n\\geq 4$.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Prove that there exists a positive integer $k$ such that $k \\\\\\\\cdot 2^n + 1$ is composite for every integer $n$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "A convex hexagon $ABCDEF$ is inscribed in a circle such that $AB=CD=EF$ and diagonals $AD,BE$, and $CF$ are concurrent. Let $P$ be the intersection of $AD$ and $CE$. Prove that $\\frac{CP}{PE}=(\\frac{AC}{CE})^2$.\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "$P$ lies between the rays $OA$ and $OB$. Find $Q$ on $OA$ and $R$ on $OB$ collinear with $P$ so that $\\frac{1}{PQ} + \\frac{1}{PR}$ is as large as possible.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Let $ABCD$ be a cyclic quadrilateral satisfying $AD^2+BC^2=AB^2$. The diagonals of $ABCD$ intersect at $E$. Let $P$ be a point on side $\\\\\\\\overline{AB}$ satisfying $\\\\\\\\angle APD=\\\\\\\\angle BPC$. Show that line $PE$ bisects $\\\\\\\\overline{CD}$.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $a$, $b$, and $c$ denote three distinct integers, and let $P$ denote a polynomial having all integral coefficients.  Show that it is impossible that $P(a)=b$, $P(b)=c$, and $P(c)=a$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "If $x, y, z$ are reals such that $0\\le x, y, z \\le 1$, show that $\\frac{x}{y + z + 1} + \\frac{y}{z + x + 1} + \\frac{z}{x + y +  1} \\le 1 - (1 - x)(1 - y)(1 - z)$\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"$a_1, a_2, \\\\\\\\ldots, a_n$ is an arbitrary sequence of positive integers. A member of the sequence is picked at \\\\nrandom. Its value is $a$. Another member is picked at random, independently of the first. Its value is $b$. Then a third value, $c$. Show that the probability that $a + b +c$ is divisible by $3$ is at least $\\\\\\\\frac14$.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "A fancy bed and breakfast inn has $5$ rooms, each with a distinctive color-coded decor.  One day $5$ friends arrive to spend the night.  There are no other guests that night.  The friends can room in any combination they wish, but with no more than $2$ friends per room.  In how many ways can the innkeeper assign the guests to the rooms?\n",
      "$\\textbf{(A) }2100\\qquad \\textbf{(B) }2220\\qquad \\textbf{(C) }3000\\qquad \\textbf{(D) }3120\\qquad \\textbf{(E) }3125\\qquad$\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $ABCD$ be a cyclic quadrilateral satisfying $AD^2+BC^2=AB^2$. The diagonals of $ABCD$ intersect at $E$. Let $P$ be a point on side $\\overline{AB}$ satisfying $\\angle APD=\\angle BPC$. Show that line $PE$ bisects $\\overline{CD}$.\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $a$, $b$, and $c$ denote three distinct integers, and let $P$ denote a polynomial having all integral coefficients.  Show that it is impossible that $P(a)=b$, $P(b)=c$, and $P(c)=a$.\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "$a_1, a_2, \\ldots, a_n$ is an arbitrary sequence of positive integers. A member of the sequence is picked at \n",
      "random. Its value is $a$. Another member is picked at random, independently of the first. Its value is $b$. Then a third value, $c$. Show that the probability that $a + b +c$ is divisible by $3$ is at least $\\frac14$.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"In triangle $ABC$, angle $A$ is twice angle $B$, angle $C$ is obtuse, and the three side lengths $a, b, c$ are integers.  Determine, with proof, the minimum possible perimeter.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"(Titu Andreescu) Let $ABCD$ be a quadrilateral circumscribed about a circle, whose interior and exterior angles are at least 60 degrees. Prove that\\\\n\\\\\\\\[\\\\\\\\frac {1}{3}|AB^3 - AD^3| \\\\\\\\le |BC^3 - CD^3| \\\\\\\\le 3|AB^3 - AD^3|.\\\\\\\\]When does equality hold?\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "(Titu Andreescu) Let $ABCD$ be a quadrilateral circumscribed about a circle, whose interior and exterior angles are at least 60 degrees. Prove that\n",
      "\\[\\frac {1}{3}|AB^3 - AD^3| \\le |BC^3 - CD^3| \\le 3|AB^3 - AD^3|.\\]\n",
      "When does equality hold?\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Problem 11\\\\nSolution\\\\nSolution 1\\\\nSolution 2\\\\nSolution 3\\\\nSolution 4\\\\nSolution 5\\\\nSolution 6\\\\nSee also\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Show that, for any fixed integer $n \\\\\\\\geq 1,$ the sequence\\\\n\\\\\\\\[2, ; 2^2, ; 2^{2^2}, ; 2^{2^{2^2}}, \\\\\\\\ldots  \\\\\\\\pmod{n}\\\\\\\\]\\\\nis eventually constant.\\\\n[The tower of exponents is defined by $a_1 = 2, ; a_{i+1} = 2^{a_i}$. Also $a_i \\\\\\\\pmod{n}$ means the remainder which results from dividing $a_i$ by $n$.]\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Prove that there are infinitely many distinct pairs $(a,b)$ of relatively prime positive integers $a>1$ and $b>1$ such that $a^b+b^a$ is divisible by $a+b$.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Let $k$ be a positive integer. Two players $A$ and $B$ play a game on an infinite grid of regular hexagons. Initially all the grid cells are empty. Then the players alternately take turns with $A$ moving first. In his move, $A$ may choose two adjacent hexagons in the grid which are empty and place a counter in both of them. In his move, $B$ may choose any counter on the board and remove it. If at any time there are $k$ consecutive grid cells in a line all of which contain a counter, $A$ wins. Find the minimum value of $k$ for which $A$ cannot win in a finite number of moves, or prove that no such minimum value exists.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Chords $AA\\'$, $BB\\'$, and $CC\\'$ of a sphere meet at an interior point $P$ but are not contained in the same plane. The sphere through $A$, $B$, $C$, and $P$ is tangent to the sphere through $A\\'$, $B\\'$, $C\\'$, and $P$. Prove that $AA\\'=BB\\'=CC\\'$.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $\\\\\\\\triangle ABC$ be an acute triangle, and let $I_B, I_C,$ and $O$ denote its $B$-excenter, $C$-excenter, and circumcenter, respectively. Points $E$ and $Y$ are selected on $\\\\\\\\overline{AC}$ such that $\\\\\\\\angle ABY = \\\\\\\\angle CBY$ and $\\\\\\\\overline{BE} \\\\\\\\perp \\\\\\\\overline{AC}.$ Similarly, points $F$ and $Z$ are selected on $\\\\\\\\overline{AB}$ such that $\\\\\\\\angle ACZ = \\\\\\\\angle BCZ$ and $\\\\\\\\overline{CF} \\\\\\\\perp \\\\\\\\overline{AB}.$ Lines $I_B F$ and $I_C E$ meet at $P.$ Prove that $\\\\\\\\overline{PO}$ and $\\\\\\\\overline{YZ}$ are perpendicular.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"For each positive integer $n$, find the number of $n$-digit positive integers that satisfy both of the following conditions:\\\\n$\\\\\\\\bullet$ no two consecutive digits are equal, and\\\\n$\\\\\\\\bullet$ the last digit is a prime.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $\\\\\\\\mathbb{Z}$ be the set of all integers. Find all pairs of integers $(a,b)$ for which there exist functions $f:\\\\\\\\mathbb{Z} \\\\\\\\rightarrow \\\\\\\\mathbb{Z}$ and $g:\\\\\\\\mathbb{Z} \\\\\\\\rightarrow \\\\\\\\mathbb{Z}$ satisfying $f(g(x)) = x+a$ and $g(f(x)) = x+b$ for all integers $x$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Two rational numbers $\\\\\\\\frac{m}{n}$ and $\\\\\\\\frac{n}{m}$ are written on a blackboard, where $m$ and $n$ are relatively prime positive integers. At any point, Evan may pick two of the numbers $x$ and $y$ written on the board and write either their arithmetic mean $\\\\\\\\frac{x+y}{2}$ or their harmonic mean $\\\\\\\\frac{2xy}{x+y}$ on the board as well. Find all pairs $(m,n)$ such that Evan can write $1$ on the board in finitely many steps.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Chords $AA'$, $BB'$, and $CC'$ of a sphere meet at an interior point $P$ but are not contained in the same plane.  The sphere through $A$, $B$, $C$, and $P$ is tangent to the sphere through $A'$, $B'$, $C'$, and $P$.  Prove that $AA'=BB'=CC'$.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $a$, $b$, $c$ be positive real numbers such that $a^2 + b^2 + c^2 + (a + b + c)^2 \\\\\\\\le 4$.  Prove that\\\\n\\\\\\\\[\\\\\\\\frac{ab + 1}{(a + b)^2} + \\\\\\\\frac{bc + 1}{(b + c)^2} + \\\\\\\\frac{ca + 1}{(c + a)^2} \\\\\\\\ge 3.\\\\\\\\]\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $\\triangle ABC$ be an acute triangle, and let $I_B, I_C,$ and $O$ denote its $B$-excenter, $C$-excenter, and circumcenter, respectively. Points $E$ and $Y$ are selected on $\\overline{AC}$ such that $\\angle ABY = \\angle CBY$ and $\\overline{BE}\\perp\\overline{AC}.$ Similarly, points $F$ and $Z$ are selected on $\\overline{AB}$ such that $\\angle ACZ = \\angle BCZ$ and $\\overline{CF}\\perp\\overline{AB}.$\n",
      "Lines $I_B F$ and $I_C E$ meet at $P.$ Prove that $\\overline{PO}$ and $\\overline{YZ}$ are perpendicular.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Karl starts with $n$ cards labeled $1,2,3,\\\\\\\\dots,n$ lined up in a random order on his desk. He calls a pair $(a,b)$ of these cards swapped if $a>b$ and the card labeled $a$ is to the left of the card labeled $b$. For instance, in the sequence of cards $3,1,4,2$, there are three swapped pairs of cards, $(3,1)$, $(3,2)$, and $(4,2)$. He picks up the card labeled 1 and inserts it back into the sequence in the opposite position: if the card labeled 1 had $i$ card to its left, then it now has $i$ cards to its right. He then picks up the card labeled 2 and reinserts it in the same manner, and so on until he has picked up and put back each of the cards $1,2,\\\\\\\\dots,n$ exactly once in that order. (For example, the process starting at $3,1,4,2$ would be $3,1,4,2\\\\\\\\to 3,4,1,2\\\\\\\\to 2,3,4,1\\\\\\\\to 2,4,3,1\\\\\\\\to 2,3,4,1$.) Show that no matter what lineup of cards Karl started with, his final lineup has the same number of swapped pairs as the starting lineup.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"(Titu Andreescu, Gabriel Dospinescu) For integral \\\\\\\\(m\\\\\\\\), let \\\\\\\\(p(m)\\\\\\\\) be the greatest prime divisor of \\\\\\\\(m\\\\\\\\). By convention, we set \\\\\\\\(p(\\\\\\\\pm 1)=1\\\\\\\\) and \\\\\\\\(p(0)=\\\\\\\\infty\\\\\\\\). Find all polynomials \\\\\\\\(f\\\\\\\\) with integer coefficients such that the sequence \\\\\\\\(\\\\\\\\{ p(f(n^2))-2n) \\\\\\\\}_{n \\\\\\\\in \\\\\\\\mathbb{Z} \\\\\\\\ge 0}\\\\\\\\) is bounded above. (In particular, this requires \\\\\\\\(f(n^2)\\\\\\\\neq 0\\\\\\\\) for \\\\\\\\(n\\\\\\\\ge 0\\\\\\\\).)\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $a$, $b$, $c$ be positive real numbers such that $a^2 + b^2 + c^2 + (a + b + c)^2 \\le 4$.  Prove that\n",
      "\\[\\frac{ab + 1}{(a + b)^2} + \\frac{bc + 1}{(b + c)^2} + \\frac{ca + 1}{(c + a)^2} \\ge 3.\\]\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Prove that there exists an infinite set of points \\\\\\\\(\\\\\\\\ldots, P_{-3}, P_{-2}, P_{-1}, P_0, P_1, P_2, P_3, \\\\\\\\ldots\\\\\\\\) in the plane with the following property: For any three distinct integers \\\\\\\\(a, b,\\\\\\\\) and \\\\\\\\(c\\\\\\\\), points \\\\\\\\(P_a\\\\\\\\), \\\\\\\\(P_b\\\\\\\\), and \\\\\\\\(P_c\\\\\\\\) are collinear if and only if \\\\\\\\(a+b+c=2014\\\\\\\\).\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"(Zuming Feng) Let $ABC$ be an acute, scalene triangle, and let $M$, $N$, and $P$ be the midpoints of $\\\\\\\\overline{BC}$, $\\\\\\\\overline{CA}$, and $\\\\\\\\overline{AB}$, respectively. Let the perpendicular bisectors of $\\\\\\\\overline{AB}$ and $\\\\\\\\overline{AC}$ intersect ray $AM$ in points $D$ and $E$ respectively, and let lines $BD$ and $CE$ intersect in point $F$, inside of triangle $ABC$. Prove that points $A$, $N$, $F$, and $P$ all lie on one circle.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Karl starts with $n$ cards labeled $1,2,3,\\dots,n$ lined up in a random order on his desk. He calls a pair $(a,b)$ of these cards swapped if $a>b$ and the card labeled $a$ is to the left of the card labeled $b$. For instance, in the sequence of cards $3,1,4,2$, there are three swapped pairs of cards, $(3,1)$, $(3,2)$, and $(4,2)$.\n",
      "He picks up the card labeled 1 and inserts it back into the sequence in the opposite position: if the card labeled 1 had $i$ card to its left, then it now has $i$ cards to its right. He then picks up the card labeled $2$ and reinserts it in the same manner, and so on until he has picked up and put back each of the cards $1,2,\\dots,n$ exactly once in that order. (For example, the process starting at $3,1,4,2$ would be $3,1,4,2\\to 3,4,1,2\\to 2,3,4,1\\to 2,4,3,1\\to 2,3,4,1$.)\n",
      "Show that no matter what lineup of cards Karl started with, his final lineup has the same number of swapped pairs as the starting lineup.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"For integer $n \\\\\\\\ge 2$, let $x_1$, $x_2$, $\\\\\\\\dots$, $x_n$ be real numbers satisfying\\\\n\\\\\\\\[x_1 + x_2 + \\\\\\\\dots + x_n = 0, \\\\\\\\quad \\\\\\\\text{and} \\\\\\\\quad x_1^2 + x_2^2 + \\\\\\\\dots + x_n^2 = 1.\\\\\\\\]\\\\nFor each subset $A \\\\\\\\subseteq \\\\\\\\{1, 2, \\\\\\\\dots, n\\\\\\\\}$, define\\\\n\\\\\\\\[S_A = \\\\\\\\sum_{i \\\\\\\\in A} x_i.\\\\\\\\]\\\\n(If $A$ is the empty set, then $S_A = 0$.)\\\\nProve that for any positive number $\\\\\\\\lambda$, the number of sets $A$ satisfying $S_A \\\\\\\\ge \\\\\\\\lambda$ is at most $2^{n - 3}/\\\\\\\\lambda^2$. For what choices of $x_1$, $x_2$, $\\\\\\\\dots$, $x_n$, $\\\\\\\\lambda$ does equality hold?\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "(Titu Andreescu, Gabriel Dospinescu) For integral $m$, let $p(m)$ be the greatest prime divisor of $m$. By convention, we set $p(\\pm 1)=1$ and $p(0)=\\infty$. Find all polynomials $f$ with integer coefficients such that the sequence $\\{ p(f(n^2))-2n) \\}_{n \\in \\mathbb{Z} \\ge 0}$ is bounded above. (In particular, this requires $f(n^2)\\neq 0$ for $n\\ge 0$.)\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Let $S$ be a set with 2002 elements, and let $N$ be an integer with $0 \\\\\\\\le N \\\\\\\\le 2^{2002}$. Prove that it is possible to color every subset of $S$ either blue or red so that the following conditions hold:\\\\n(a) the union of any two red subsets is red;\\\\n(b) the union of any two blue subsets is blue;\\\\n(c) there are exactly $N$ red subsets.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "(Zuming Feng) Let $ABC$ be an acute, [scalene](https://artofproblemsolving.com/wiki/index.php/Scalene) triangle, and let $M$, $N$, and $P$ be the midpoints of $\\overline{BC}$, $\\overline{CA}$, and $\\overline{AB}$, respectively. Let the [perpendicular](https://artofproblemsolving.com/wiki/index.php/Perpendicular) [bisectors](https://artofproblemsolving.com/wiki/index.php/Bisect) of $\\overline{AB}$ and $\\overline{AC}$ intersect ray $AM$ in points $D$ and $E$ respectively, and let lines $BD$ and $CE$ intersect in point $F$, inside of triangle $ABC$. Prove that points $A$, $N$, $F$, and $P$ all lie on one circle.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $u$ and $v$ be real numbers such that\\\\n\\\\\\\\[(u + u^2 + u^3 + \\\\\\\\cdots + u^8) + 10u^9 = (v + v^2 + v^3 + \\\\\\\\cdots + v^{10}) + 10v^{11} = 8.\\\\\\\\]Determine, with proof, which of the two numbers, $u$ or $v$, is larger.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "For integer $n \\ge 2$, let $x_1$, $x_2$, $\\dots$, $x_n$ be real numbers satisfying\n",
      "\\[x_1 + x_2 + \\dots + x_n = 0, \\quad \\text{and} \\quad x_1^2 + x_2^2 + \\dots + x_n^2 = 1.\\]\n",
      "For each subset $A \\subseteq \\{1, 2, \\dots, n\\}$, define\n",
      "\\[S_A = \\sum_{i \\in A} x_i.\\]\n",
      "(If $A$ is the empty set, then $S_A = 0$.)\n",
      "Prove that for any positive number $\\lambda$, the number of sets $A$ satisfying $S_A \\ge \\lambda$ is at most $2^{n - 3}/\\lambda^2$.  For what choices of $x_1$, $x_2$, $\\dots$, $x_n$, $\\lambda$ does equality hold?\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"If $P(x)$, $Q(x)$, $R(x)$, and $S(x)$ are all polynomials such that\\\\n\\\\\\\\[P(x^5) + xQ(x^5) + x^2 R(x^5) = (x^4 + x^3 + x^2 + x +1) S(x),\\\\\\\\]\\\\nprove that $x-1$ is a factor of $P(x)$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $S$ be a set with 2002 elements, and let $N$ be an integer with $0 \\le N \\le 2^{2002}$.  Prove that it is possible to color every subset of $S$ either blue or red so that the following conditions hold:\n",
      "(a) the union of any two red subsets is red;\n",
      "(b) the union of any two blue subsets is blue;\n",
      "(c) there are exactly $N$ red subsets.\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Triangle $AMC$ is isosceles with $AM = AC$. Medians $\\overline{MV}$ and $\\overline{CU}$ are perpendicular to each other, and $MV=CU=12$. What is the area of $\\triangle AMC?$\n",
      "\n",
      "$\\textbf{(A) } 48 \\qquad \\textbf{(B) } 72 \\qquad \\textbf{(C) } 96 \\qquad \\textbf{(D) } 144 \\qquad \\textbf{(E) } 192$\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "If $P(x)$, $Q(x)$, $R(x)$, and $S(x)$ are all [polynomials](https://artofproblemsolving.com/wiki/index.php/Polynomial) such that\n",
      "\\[P(x^5) + xQ(x^5) + x^2 R(x^5) = (x^4 + x^3 + x^2 + x +1) S(x),\\]\n",
      "prove that $x-1$ is a factor of $P(x)$.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"The 20 members of a local tennis club have scheduled exactly 14 two-person games among themselves, with each member playing in at least one game. Prove that within this schedule there must be a set of 6 games with 12 distinct players.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Joey and Chloe and their daughter Zoe all have the same birthday. Joey is $1$ year older than Chloe, and Zoe is exactly $1$ year old today. Today is the first of the $9$ birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?\n",
      "$\\textbf{(A) }7 \\qquad \\textbf{(B) }8 \\qquad \\textbf{(C) }9 \\qquad \\textbf{(D) }10 \\qquad \\textbf{(E) }11 \\qquad$\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "The 20 members of a local tennis club have scheduled exactly 14 two-person games among themselves, with each member playing in at least one game. Prove that within this schedule there must be a set of 6 games with 12 distinct players.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $a$, $b$, $c$ be real numbers greater than or equal to $1$. Prove that $\\\\\\\\min{\\\\\\\\left (\\\\\\\\frac{10a^2-5a+1}{b^2-5b+10},\\\\\\\\frac{10b^2-5b+1}{c^2-5c+10},\\\\\\\\frac{10c^2-5c+1}{a^2-5a+10}\\\\\\\\right )} \\\\\\\\leq abc$\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"An equilateral triangle $\\\\\\\\Delta$ of side length $L>0$ is given. Suppose that $n$ equilateral triangles with side length 1 and with non-overlapping interiors are drawn inside $\\\\\\\\Delta$, such that each unit equilateral triangle has sides parallel to $\\\\\\\\Delta$, but with opposite orientation. (An example with $n=2$ is drawn below.)\\\\n[asy] draw((0,0)--(1,0)--(1/2,sqrt(3)/2)--cycle,linewidth(0.5)); filldraw((0.45,0.55)--(0.65,0.55)--(0.55,0.55-sqrt(3)/2*0.2)--cycle,gray,linewidth(0.5)); filldraw((0.54,0.3)--(0.34,0.3)--(0.44,0.3-sqrt(3)/2*0.2)--cycle,gray,linewidth(0.5)); [/asy]\\\\nProve that\\\\\\\\[n \\\\\\\\leq \\\\\\\\frac{2}{3} L^{2}.\\\\\\\\]\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Find all solutions to $(m^2+n)(m + n^2)= (m - n)^3$, where m and n are non-zero integers.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"If a point $A_1$ is in the interior of an equilateral triangle $ABC$ and point $A_2$ is in the interior of $\\\\\\\\triangle{A_1BC}$, prove that\\\\n$I.Q. (A_1BC) > I.Q.(A_2BC)$, \\\\nwhere the isoperimetric quotient of a figure $F$ is defined by \\\\n$I.Q.(F) = \\\\\\\\frac{\\\\\\\\text{Area (F)}}{\\\\\\\\text{[Perimeter (F)]}^2}$\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Let $O$ and $H$ be the circumcenter and the orthocenter of an acute triangle $ABC$. Points $M$ and $D$ lie on side $BC$ such that $BM = CM$ and $\\\\\\\\angle BAD = \\\\\\\\angle CAD$. Ray $MO$ intersects the circumcircle of triangle $BHC$ in point $N$. Prove that $\\\\\\\\angle ADO = \\\\\\\\angle HAN$.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSINGRetrying answer fetching:\n",
      "'```json\\n{\\n\"problem\": \"(Răzvan Gelca) Prove that the system \\\\\\\\begin{align*}x^6 + x^3 + x^3y + y &= 147^{157} \\\\\\\\\\\\\\\\ x^3 + x^3y + y^2 + y + z^9 &= 157^{147}\\\\\\\\end{align*} has no solutions in integers $x$, $y$, and $z$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "An equilateral triangle $\\Delta$ of side length $L>0$ is given. Suppose that $n$ equilateral triangles with side length 1 and with non-overlapping interiors are drawn inside $\\Delta$, such that each unit equilateral triangle has sides parallel to $\\Delta$, but with opposite orientation. (An example with $n=2$ is drawn below.)\n",
      "[asy] draw((0,0)--(1,0)--(1/2,sqrt(3)/2)--cycle,linewidth(0.5)); filldraw((0.45,0.55)--(0.65,0.55)--(0.55,0.55-sqrt(3)/2*0.2)--cycle,gray,linewidth(0.5)); filldraw((0.54,0.3)--(0.34,0.3)--(0.44,0.3-sqrt(3)/2*0.2)--cycle,gray,linewidth(0.5)); [/asy]\n",
      "Prove that\\[n \\leq \\frac{2}{3} L^{2}.\\]\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "If a point $A_1$ is in the interior of an equilateral triangle $ABC$ and point $A_2$ is in the interior of $\\triangle{A_1BC}$, prove that\n",
      "$I.Q. (A_1BC) > I.Q.(A_2BC)$,\n",
      "where the isoperimetric quotient of a figure $F$ is defined by\n",
      "$I.Q.(F) = \\frac{\\text{Area (F)}}{\\text{[Perimeter (F)]}^2}$\n",
      "Retrying answer fetching:\n",
      "A\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"An integer $n$ will be called good if we can write\\\\n$n=a_1+a_2+\\\\\\\\cdots+a_k$,\\\\nwhere $a_1,a_2, \\\\\\\\ldots, a_k$ are positive integers (not necessarily distinct) satisfying\\\\n$\\\\\\\\frac{1}{a_1}+\\\\\\\\frac{1}{a_2}+\\\\\\\\cdots+\\\\\\\\frac{1}{a_k}=1$.\\\\nGiven the information that the integers 33 through 73 are good, prove that every integer $\\\\\\\\ge 33$ is good.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "A large [equilateral triangle](https://artofproblemsolving.com/wiki/index.php/Equilateral_triangle) is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure, we have $3$ rows of small congruent equilateral triangles, with $5$ small triangles in the base row.  How many toothpicks would be needed to construct a large equilateral triangle if the base row of the triangle consists of $2003$ small equilateral triangles? \n",
      "\n",
      "$\\mathrm{(A) \\ } 1,004,004 \\qquad \\mathrm{(B) \\ } 1,005,006 \\qquad \\mathrm{(C) \\ } 1,507,509 \\qquad \\mathrm{(D) \\ } 3,015,018 \\qquad \\mathrm{(E) \\ } 6,021,018$\n",
      "Gemini not happy.\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "An integer $n$ will be called good if we can write\n",
      "$n=a_1+a_2+\\cdots+a_k$,\n",
      "where $a_1,a_2, \\ldots, a_k$ are positive integers (not necessarily distinct) satisfying\n",
      "$\\frac{1}{a_1}+\\frac{1}{a_2}+\\cdots+\\frac{1}{a_k}=1$.\n",
      "Given the information that the integers 33 through 73 are good, prove that every integer $\\ge 33$ is good.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"An empty $2020 \\\\\\\\times 2020 \\\\\\\\times 2020$ cube is given, and a $2020 \\\\\\\\times 2020$ grid of square unit cells is drawn  on each of its six faces. A beam is a $1 \\\\\\\\times 1 \\\\\\\\times 2020$ rectangular prism. Several beams are placed inside the cube subject to the following conditions:\\\\n- The two $1 \\\\\\\\times 1$ faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are $3 \\\\\\\\cdot {2020}^2$ possible positions for a beam.)\\\\n- No two beams have intersecting interiors.\\\\n- The interiors of each of the four $1 \\\\\\\\times 2020$ faces of each beam touch either a face of the cube or the interior of the face of another beam.\\\\nWhat is the smallest positive number of beams that can be placed to satisfy these conditions?\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Determine all non-negative integral solutions $(n_1,n_2,\\\\\\\\dots , n_{14})$ if any, apart from permutations, of the Diophantine Equation $n_1^4+n_2^4+\\\\\\\\cdots +n_{14}^4=1599$.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Let $R$ denote a non-negative rational number. Determine a fixed set of integers $a,b,c,d,e,f$, such that for every choice of $R$,\\\\n\\\\n$\\\\\\\\left|\\\\\\\\frac{aR^2+bR+c}{dR^2+eR+f}-\\\\\\\\sqrt[3]{2}\\\\\\\\right|<|R-\\\\\\\\sqrt[3]{2}|$\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $R$ denote a non-negative rational number. Determine a fixed set of integers $a,b,c,d,e,f$, such that for every choice of $R$, \n",
      "\n",
      "$\\left|\\frac{aR^2+bR+c}{dR^2+eR+f}-\\sqrt[3]{2}\\right|<|R-\\sqrt[3]{2}|$\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"$\\\\\\\\triangle ABC$ is a triangle. Take points $D, E, F$ on the perpendicular bisectors of $BC, CA, AB$ respectively. Show that the lines through $A, B, C$ perpendicular to $EF, FD, DE$ respectively are concurrent.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Find the maximum possible number of three term arithmetic progressions in a monotone sequence of $n$ distinct reals.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "$\\triangle ABC$ is a triangle. Take points $D, E, F$ on the perpendicular bisectors of $BC, CA, AB$ respectively. Show that the lines through $A, B, C$ perpendicular to $EF, FD, DE$ respectively are concurrent.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"(Titu Andreescu)\\\\nLet $a$, $b$, and $c$ be positive real numbers.  Prove that\\\\n\\\\n$(a^5 - a^2 + 3)(b^5 - b^2 + 3)(c^5 - c^2 + 3) \\\\\\\\ge (a+b+c)^3$.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"What is the smallest integer $n$, greater than one, for which the root-mean-square of the first $n$ positive integers is an integer?\\\\n$\\\\\\\\mathbf{Note.}$ The root-mean-square of $n$ numbers $a_1, a_2, \\\\\\\\cdots, a_n$ is defined to be\\\\n$\\\\\\\\left[\\\\\\\\frac{a_1^2 + a_2^2 + \\\\\\\\cdots + a_n^2}n\\\\\\\\right]^{1/2}$\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "(Titu Andreescu)\n",
      "Let $a$, $b$, and $c$ be positive real numbers.  Prove that\n",
      "\n",
      "\n",
      "$(a^5 - a^2 + 3)(b^5 - b^2 + 3)(c^5 - c^2 + 3) \\ge (a+b+c)^3$.\n",
      "Gemini not happy.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $\\\\\\\\mathbb{Z}$ be the set of integers. Find all functions $f : \\\\\\\\mathbb{Z} \\\\\\\\rightarrow \\\\\\\\mathbb{Z}$ such that \\\\\\\\[xf(2f(y)-x)+y^2f(2x-f(y))=\\\\\\\\frac{f(x)^2}{x}+f(yf(y))\\\\\\\\] for all $x, y \\\\\\\\in \\\\\\\\mathbb{Z}$ with $x \\\\\\\\neq 0$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"In 1960 only 5% of the working adults in Carlin City worked\\\\nat home. By 1970 the \\\\\"at-home\\\\\" work force had increased to\\\\n8%. In 1980 there were approximately 15% working at home,\\\\nand in 1990 there were 30%. The graph that best illustrates\\\\nthis is:\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"A given tetrahedron $ABCD$ is isosceles, that is, $AB=CD, AC=BD, AD=BC$. Show that the faces of the tetrahedron are acute-angled triangles.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $A,B,C,D$ denote four points in space such that at most one of the distances $AB,AC,AD,BC,BD,CD$ is greater than $1$. Determine the maximum value of the sum of the six distances.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"An angle is drawn on a set of equally spaced parallel lines as shown. The ratio of the area of shaded region $C$ to the area of shaded region $B$ is 11/5. Find the ratio of shaded region $D$ to the area of shaded region $A.$\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Let $ABC$ be a triangle. A circle passing through $A$ and $B$ intersects segments $AC$ and $BC$ at $D$ and $E$, respectively. Lines $AB$ and $DE$ intersect at $F$, while lines $BD$ and $CF$ intersect at $M$. Prove that $MF = MC$ if and only if $MB \\\\\\\\cdot MD = MC^2$.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $a_0, a_1, a_2,\\\\\\\\cdots$ be a sequence of positive real numbers satisfying $a_{i-1}a_{i+1}\\\\\\\\le a^2_i$\\\\nfor $i = 1, 2, 3,\\\\\\\\cdots$ . (Such a sequence is said to be log concave.) Show that for\\\\neach $n > 1$,\\\\n\\\\\\\\[\\\\\\\\frac{a_0+\\\\\\\\cdots+a_n}{n+1}\\\\\\\\cdot\\\\\\\\frac{a_1+\\\\\\\\cdots+a_{n-1}}{n-1}\\\\\\\\ge\\\\\\\\frac{a_0+\\\\\\\\cdots+a_{n-1}}{n}\\\\\\\\cdot\\\\\\\\frac{a_1+\\\\\\\\cdots+a_{n}}{n}.\\\\\\\\]\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $ABC$ be a triangle. A circle passing through $A$ and $B$ intersects segments $AC$ and $BC$ at $D$ and $E$, respectively. Lines $AB$ and $DE$ intersect at $F$, while lines $BD$ and $CF$ intersect at $M$. Prove that $MF = MC$ if and only if $MB\\cdot MD = MC^2$.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $ABCD$ be a cyclic quadrilateral with $AB=4, BC=5, CD=6,$ and $DA=7.$ Let $A_1$ and $C_1$ be the feet of the perpendiculars from $A$ and $C,$ respectively, to line $BD,$ and let $B_1$ and $D_1$ be the feet of the perpendiculars from $B$ and $D,$ respectively, to line $AC.$ The perimeter of $A_1B_1C_1D_1$ is $\\\\\\\\frac mn,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $a_0, a_1, a_2,\\cdots$ be a sequence of positive real numbers satisfying $a_{i-1}a_{i+1}\\le a^2_i$\n",
      "for $i = 1, 2, 3,\\cdots$ . (Such a sequence is said to be log concave.) Show that for\n",
      "each $n > 1$,\n",
      "\\[\\frac{a_0+\\cdots+a_n}{n+1}\\cdot\\frac{a_1+\\cdots+a_{n-1}}{n-1}\\ge\\frac{a_0+\\cdots+a_{n-1}}{n}\\cdot\\frac{a_1+\\cdots+a_{n}}{n}.\\]\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $P(z) = z^n + c_1 z^{n-1} + c_2 z^{n-2} + \\\\\\\\cdots + c_n$ be a polynomial in the complex variable $z$, with real coefficients $c_k$. Suppose that $|P(i)| < 1$. Prove that there exist real numbers $a$ and $b$ such that $P(a + bi) = 0$ and $(a^2 + b^2 + 1)^2 < 4 b^2 + 1$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Rabbits Peter and Pauline have three offspring—Flopsie, Mopsie, and Cotton-tail. These five rabbits are to be distributed to four different pet stores so that no store gets both a parent and a child. It is not required that every store gets a rabbit. In how many different ways can this be done?\n",
      "$\\textbf{(A)} \\ 96 \\qquad  \\textbf{(B)} \\ 108 \\qquad  \\textbf{(C)} \\ 156 \\qquad  \\textbf{(D)} \\ 204 \\qquad  \\textbf{(E)} \\ 372$\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $\\\\\\\\omega$ be the incircle of a fixed equilateral triangle $ABC$. Let $\\\\\\\\ell$ be a variable line that is tangent to $\\\\\\\\omega$ and meets the interior of segments $BC$ and $CA$ at points $P$ and $Q$, respectively. A point $R$ is chosen such that $PR = PA$ and $QR = QB$. Find all possible locations of the point $R$, over all choices of $\\\\\\\\ell$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $P(z)= z^n + c_1 z^{n-1} + c_2 z^{n-2} + \\cdots + c_n$ be a [polynomial](https://artofproblemsolving.com/wiki/index.php/Polynomial) in the complex variable $z$, with real coefficients $c_k$. Suppose that $|P(i)| < 1$. Prove that there exist real numbers $a$ and $b$ such that $P(a + bi) = 0$ and $(a^2 + b^2 + 1)^2 < 4 b^2 + 1$.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"A deck of $n$ playing cards, which contains three aces, is shuffled at random (it is assumed that all possible card distributions are equally likely). The cards are then turned up one by one from the top until the second ace appears. Prove that the expected (average) number of cards to be turned up is $(n+1)/2$.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"There are $a+b$ bowls arranged in a row, numbered $1$ through $a+b$, where $a$ and $b$ are given positive integers. Initially, each of the first $a$ bowls contains an apple, and each of the last $b$ bowls contains a pear. A legal move consists of moving an apple from bowl $i$ to bowl $i+1$ and a pear from bowl $j$ to bowl $j-1$, provided that the difference $i-j$ is even. We permit multiple fruits in the same bowl at the same time. The goal is to end up with the first $b$ bowls each containing a pear and the last $a$ bowls each containing an apple. Show that this is possible if and only if the product $ab$ is even.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Point $A,B,C,D,$ and $E$ are equally spaced on a minor arc of a circle. Points $E,F,G,H,I$ and $A$ are equally spaced on a minor arc of a second circle with center $C$ as shown in the figure below. The angle $\\\\\\\\angle ABD$ exceeds $\\\\\\\\angle AHG$ by $12^\\\\\\\\circ$. Find the degree measure of $\\\\\\\\angle BAG$.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $a_1, b_1, a_2, b_2, \\\\\\\\dots , a_n, b_n$ be nonnegative real numbers. Prove that\\\\n\\\\\\\\[\\\\\\\\sum_{i, j = 1}^{n} \\\\\\\\min\\\\\\\\{a_ia_j, b_ib_j\\\\\\\\} \\\\\\\\le \\\\\\\\sum_{i, j = 1}^{n} \\\\\\\\min\\\\\\\\{a_ib_j, a_jb_i\\\\\\\\}.\\\\\\\\]\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "A deck of $n$ playing cards, which contains three aces, is shuffled at random (it is assumed that all possible card distributions are equally likely). The cards are then turned up one by one from the top until the second ace appears. Prove that the expected (average) number of cards to be turned up is $(n+1)/2$.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $ABC$ be a triangle with $\\\\\\\\angle A = 90^{\\\\\\\\circ}$. Points $D$ and $E$ lie on sides $AC$ and $AB$, respectively, such that $\\\\\\\\angle ABD = \\\\\\\\angle DBC$ and $\\\\\\\\angle ACE = \\\\\\\\angle ECB$. Segments $BD$ and $CE$ meet at $I$.  Determine whether or not it is possible for segments $AB, AC, BI, ID, CI, IE$ to all have integer lengths.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"(Gabriel Carroll) Let $n$ be a positive integer. Denote by $S_n$ the set of points $(x, y)$ with integer coordinates such that\\\\n\\\\\\\\[\\\\\\\\left|x\\\\\\\\right| + \\\\\\\\left|y + \\\\\\\\frac {1}{2}\\\\\\\\right| < n\\\\\\\\]\\\\nA path is a sequence of distinct points $(x_1 , y_1 ), (x_2 , y_2 ), \\\\\\\\ldots , (x_\\\\\\\\ell, y_\\\\\\\\ell)$ in $S_n$ such that, for $i = 2, \\\\\\\\ldots , \\\\\\\\ell$, the distance between $(x_i , y_i )$ and $(x_{i - 1} , y_{i - 1} )$ is $1$ (in other words, the points $(x_i , y_i )$ and $(x_{i - 1} , y_{i - 1} )$ are neighbors in the lattice of points with integer coordinates). Prove that the points in $S_n$ cannot be partitioned into fewer than $n$ paths (a partition of $S_n$ into $m$ paths is a set $\\\\\\\\mathcal{P}$ of $m$ nonempty paths such that each point in $S_n$ appears in exactly one of the $m$ paths in $\\\\\\\\mathcal{P}$).\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $a_1, b_1, a_2, b_2, \\dots , a_n, b_n$ be nonnegative real numbers. Prove that\n",
      "\\[\\sum_{i, j = 1}^{n} \\min\\{a_ia_j, b_ib_j\\} \\le \\sum_{i, j = 1}^{n} \\min\\{a_ib_j, a_jb_i\\}.\\]\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"The cubic polynomial $x^3+ax^2+bx+c$ has real coefficients and three real roots $r \\\\\\\\ge s \\\\\\\\ge t$. Show that $k=a^2-3b \\\\\\\\ge 0$ and that $\\\\\\\\sqrt k \\\\\\\\le r-t$.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Susan had 50 dollars to spend at the carnival. She spent 12 dollars on food and twice as much on rides. How many dollars did she have left to spend?\",\\n\"A\": \"12\",\\n\"B\": \"14\",\\n\"C\": \"16\",\\n\"D\": \"26\",\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"The 2010 positive numbers $a_1, a_2, \\\\\\\\dots , a_{2010}$ satisfy\\\\nthe inequality $a_ia_j \\\\\\\\le i+j$ for all distinct indices $i, j$.\\\\nDetermine, with proof, the largest possible value of the product\\\\n$a_1a_2\\\\\\\\cdots a_{2010}$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $n > 1$ be an integer. Find, with proof, all sequences $x_1, x_2, \\\\\\\\ldots, x_{n-1}$ of positive integers with the following three properties:\\\\n\\\\n(a) $x_1 < x_2 < \\\\\\\\cdots < x_{n-1}$;\\\\n(b) $x_i + x_{n-i} = 2n$ for all $i=1,2,\\\\\\\\ldots,n-1$;\\\\n(c) given any two indices $i$ and $j$ (not necessarily distinct) for which $x_i + x_j < 2n$, there is an index $k$ such that $x_i + x_j = x_k$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $ABCD$ be a tetrahedron with $AB=41$, $AC=7$, $AD=18$, $BC=36$, $BD=27$, and $CD=13$. Let $d$ be the distance between the midpoints of edges $AB$ and $CD$. Find $d^{2}$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Let $P_1$, $P_2$, $\\\\\\\\dots$, $P_{2n}$ be $2n$ distinct points on the unit circle $x^2+y^2=1$, other than $(1,0)$. Each point is colored either red or blue, with exactly $n$ red points and $n$ blue points. Let $R_1$, $R_2$, $\\\\\\\\dots$, $R_n$ be any ordering of the red points. Let $B_1$ be the nearest blue point to $R_1$ traveling counterclockwise around the circle starting from $R_1$. Then let $B_2$ be the nearest of the remaining blue points to $R_2$ travelling counterclockwise around the circle from $R_2$, and so on, until we have labeled all of the blue points $B_1, \\\\\\\\dots, B_n$. Show that the number of counterclockwise arcs of the form $R_i \\\\\\\\to B_i$ that contain the point $(1,0)$ is independent of the way we chose the ordering $R_1, \\\\\\\\dots, R_n$ of the red points.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Find, with proof, all positive integers $n$ for which $2^n + 12^n + 2011^n$ is a perfect square.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Consider functions $f : [0, 1] \\\\\\\\rightarrow \\\\\\\\mathbb{R}$ which satisfy\\\\n\\\\n(i) $f(x) \\\\\\\\ge 0$ for all $x$ in $[0, 1]$,\\\\n\\\\n(ii) $f(1) = 1$,\\\\n\\\\n(iii) $f(x) + f(y) \\\\\\\\le f(x + y)$ whenever $x$, $y$, and $x + y$ are all in $[0, 1]$.\\\\nFind, with proof, the smallest constant $c$ such that\\\\n\\\\n$f(x) \\\\\\\\le cx$\\\\nfor every function $f$ satisfying (i)-(iii) and every $x$ in $[0, 1]$.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Seven distinct pieces of candy are to be distributed among three bags. The red bag and the blue bag must each receive at least one piece of candy; the white bag may remain empty. How many arrangements are possible?\n",
      "$\\textbf{(A)}\\ 1930 \\qquad \\textbf{(B)}\\ 1931 \\qquad \\textbf{(C)}\\ 1932 \\qquad \\textbf{(D)}\\ 1933 \\qquad \\textbf{(E)}\\ 1934$\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $P_1$, $P_2$, $\\dots$, $P_{2n}$ be $2n$ distinct points on the unit circle $x^2+y^2=1$, other than $(1,0)$. Each point is colored either red or blue, with exactly $n$ red points and $n$ blue points. Let $R_1$, $R_2$, $\\dots$, $R_n$ be any ordering of the red points. Let $B_1$ be the nearest blue point to $R_1$ traveling counterclockwise around the circle starting from $R_1$. Then let $B_2$ be the nearest of the remaining blue points to $R_2$ travelling counterclockwise around the circle from $R_2$, and so on, until we have labeled all of the blue points $B_1, \\dots, B_n$. Show that the number of counterclockwise arcs of the form $R_i \\to B_i$ that contain the point $(1,0)$ is independent of the way we chose the ordering $R_1, \\dots, R_n$ of the red points.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Given a triangle $ABC$, let $P$ and $Q$ be points on segments $\\\\\\\\overline{AB}$ and $\\\\\\\\overline{AC}$, respectively, such that $AP = AQ$. Let $S$ and $R$ be distinct points on segment $\\\\\\\\overline{BC}$ such that $S$ lies between $B$ and $R$, $\\\\\\\\angle BPS = \\\\\\\\angle PRS$, and $\\\\\\\\angle CQR = \\\\\\\\angle QSR$. Prove that $P$, $Q$, $R$, $S$ are concyclic (in other words, these four points lie on a circle).\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "After simple interest for two months at $5$% per annum was credited, a Boy Scout Troop had a total \n",
      "of $\\textdollar 255.31$ in the Council Treasury. The interest credited was a number of dollars plus the following number of cents\n",
      "$\\textbf{(A) }11\\qquad \\textbf{(B) }12\\qquad \\textbf{(C) }13\\qquad \\textbf{(D) }21\\qquad  \\textbf{(E) }31$\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Given a triangle $ABC$, let $P$ and $Q$ be points on segments $\\overline{AB}$ and $\\overline{AC}$, respectively, such that $AP = AQ$.  Let $S$ and $R$ be distinct points on segment $\\overline{BC}$ such that $S$ lies between $B$ and $R$, $\\angle{BPS} = \\angle{PRS}$, and $\\angle{CQR} = \\angle{QSR}$.  Prove that $P$, $Q$, $R$, $S$ are concyclic (in other words, these four points lie on a circle).\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Three identical square sheets of paper each with side length $6$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\\circ$ about its center and the top sheet is rotated clockwise $60^\\circ$ about its center, resulting in the $24$-sided polygon shown in the figure below. The area of this polygon can be expressed in the form $a-b\\sqrt{c}$, where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime. What is $a+b+c$?\n",
      "\n",
      "[asy] defaultpen(fontsize(8)+0.8); size(150); pair O,A1,B1,C1,A2,B2,C2,A3,B3,C3,A4,B4,C4; real x=45, y=90, z=60; O=origin;  A1=dir(x); A2=dir(x+y); A3=dir(x+2y); A4=dir(x+3y); B1=dir(x-z); B2=dir(x+y-z); B3=dir(x+2y-z); B4=dir(x+3y-z); C1=dir(x-2z); C2=dir(x+y-2z); C3=dir(x+2y-2z); C4=dir(x+3y-2z); draw(A1--A2--A3--A4--A1, gray+0.25+dashed); filldraw(B1--B2--B3--B4--cycle, white, gray+dashed+linewidth(0.25)); filldraw(C1--C2--C3--C4--cycle, white, gray+dashed+linewidth(0.25)); dot(O); pair P1,P2,P3,P4,Q1,Q2,Q3,Q4,R1,R2,R3,R4; P1=extension(A1,A2,B1,B2); Q1=extension(A1,A2,C3,C4);  P2=extension(A2,A3,B2,B3); Q2=extension(A2,A3,C4,C1);  P3=extension(A3,A4,B3,B4); Q3=extension(A3,A4,C1,C2);  P4=extension(A4,A1,B4,B1); Q4=extension(A4,A1,C2,C3);  R1=extension(C2,C3,B2,B3); R2=extension(C3,C4,B3,B4);  R3=extension(C4,C1,B4,B1); R4=extension(C1,C2,B1,B2); draw(A1--P1--B2--R1--C3--Q1--A2); draw(A2--P2--B3--R2--C4--Q2--A3); draw(A3--P3--B4--R3--C1--Q3--A4); draw(A4--P4--B1--R4--C2--Q4--A1); [/asy]\n",
      "$(\\textbf{A})\\: 75\\qquad(\\textbf{B}) \\: 93\\qquad(\\textbf{C}) \\: 96\\qquad(\\textbf{D}) \\: 129\\qquad(\\textbf{E}) \\: 147$\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Given a nonisosceles, nonright triangle $ABC,$ let $O$ denote the center of its circumscribed circle, and let $A_1, \\\\\\\\, B_1,$ and $C_1$ be the midpoints of sides $BC, \\\\\\\\, CA,$ and $AB,$ respectively.  Point $A_2$ is located on the ray $OA_1$ so that $\\\\\\\\triangle OAA_1$ is similar to $\\\\\\\\triangle OA_2A$.  Points $B_2$ and $C_2$ on rays $OB_1$ and $OC_1,$ respectively, are defined similarly.  Prove that lines $AA_2, \\\\\\\\, BB_2,$ and $CC_2$ are concurrent, i.e. these three lines intersect at a point.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "The point $P(a,b)$ in the $xy$-plane is first rotated counterclockwise by $90^\\circ$ around the point $(1,5)$ and then reflected about the line $y = -x$. The image of $P$ after these two transformations is at $(-6,3)$. What is $b - a ?$\n",
      "$\\textbf{(A)} ~1 \\qquad\\textbf{(B)} ~3 \\qquad\\textbf{(C)} ~5 \\qquad\\textbf{(D)} ~7 \\qquad\\textbf{(E)} ~9$\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"(Kiran Kedlaya) Let $n$ be an integer greater than 1. Suppose $2n$ points are given in the plane, no three of which are collinear. Suppose $n$ of the given $2n$ points are colored blue and the other $n$ colored red. A line in the plane is called a balancing line if it passes through one blue and one red point and, for each side of the line, the number of blue points on that side is equal to the number of red points on the same side.\\\\nProve that there exist at least two balancing lines.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $\\\\\\\\, |U|, \\\\\\\\, \\\\\\\\sigma(U) \\\\\\\\,$ and $\\\\\\\\, \\\\\\\\pi(U) \\\\\\\\,$ denote the number of elements, the sum, and the product, respectively, of a finite set $\\\\\\\\, U \\\\\\\\,$ of positive integers.  (If $\\\\\\\\, U \\\\\\\\,$ is the empty set, $\\\\\\\\, |U| = 0, \\\\\\\\, \\\\\\\\sigma(U) = 0, \\\\\\\\, \\\\\\\\pi(U) = 1$.) Let $\\\\\\\\, S \\\\\\\\,$ be a finite set of positive integers. As usual, let $\\\\\\\\, \\\\\\\\binom{n}{k} \\\\\\\\,$ denote $\\\\\\\\, \\\\\\\\frac{n!}{k! (n-k)!}$. Prove that $\\\\\\\\sum_{U \\\\\\\\subseteq S} (-1)^{|U|} \\\\\\\\binom{m - \\\\\\\\sigma(U)}{|S|} = \\\\\\\\pi(S)$ for all integers $\\\\\\\\, m \\\\\\\\geq \\\\\\\\sigma(S)$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Let $\\\\\\\\mathbf{Z}$ denote the set of all integers. Find all real numbers $c > 0$ such that there exists a labeling of the lattice points $( x, y ) \\\\\\\\in \\\\\\\\mathbf{Z}^2$ with positive integers for which: only finitely many distinct labels occur, and for each label $i$, the distance between any two points labeled $i$ is at least $c^i$.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $\\, |U|, \\, \\sigma(U) \\,$ and $\\, \\pi(U) \\,$ denote the number of elements, the sum, and the product, respectively, of a finite set $\\, U \\,$ of positive integers.  (If $\\, U \\,$ is the empty set, $\\, |U| = 0, \\, \\sigma(U) = 0, \\, \\pi(U) = 1$.) Let $\\, S \\,$ be a finite set of positive integers. As usual, let $\\, \\binom{n}{k} \\,$ denote $\\, n! \\over k! \\, (n-k)!$. Prove that \\[\\sum_{U \\subseteq S} (-1)^{|U|} \\binom{m - \\sigma(U)}{|S|} = \\pi(S)\\] for all integers $\\, m \\geq \\sigma(S)$.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $S$ be the set of all triangles $ABC$ for which\\\\n\\\\\\\\[5 \\\\\\\\left( \\\\\\\\dfrac{1}{AP} + \\\\\\\\dfrac{1}{BQ} + \\\\\\\\dfrac{1}{CR} \\\\\\\\right) - \\\\\\\\dfrac{3}{\\\\\\\\min\\\\\\\\{ AP, BQ, CR \\\\\\\\}} = \\\\\\\\dfrac{6}{r},\\\\\\\\]where $r$ is the inradius and $P, Q, R$ are the points of tangency of the incircle with sides $AB, BC, CA,$ respectively. Prove that all triangles in $S$ are isosceles  and similar to one another.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"If $A$ and $B$ are fixed points on a given circle and $XY$ is a variable diameter of the same circle, determine the locus of the point of intersection of lines $AX$ and $BY$. You may assume that $AB$ is not a diameter.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "In $\\triangle ABC, AB = 10~ AC = 8$ and $BC = 6$. Circle $P$ is the circle with smallest radius which passes through $C$ and is tangent to $AB$. Let $Q$ and $R$ be the points of intersection, distinct from $C$ , of circle $P$ with sides $AC$ and $BC$, respectively. The length of segment $QR$ is\n",
      "$\\textbf{(A) }4.75\\qquad \\textbf{(B) }4.8\\qquad \\textbf{(C) }5\\qquad \\textbf{(D) }4\\sqrt{2}\\qquad  \\textbf{(E) }3\\sqrt{3}$\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $a$, $b$, $c$ be positive real numbers. Prove that\\\\n\\\\n$\\\\\\\\dfrac{(2a + b + c)^2}{2a^2 + (b + c)^2} + \\\\\\\\dfrac{(2b + c + a)^2}{2b^2 + (c + a)^2} + \\\\\\\\dfrac{(2c + a + b)^2}{2c^2 + (a + b)^2} \\\\\\\\le 8.$\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"A permutation of the set of positive integers $\\\\\\\\{1, 2, \\\\\\\\ldots, n\\\\\\\\}$ is a sequence $(a_1, a_2, \\\\\\\\ldots, a_n)$ such that each element of $\\\\\\\\{1, 2, \\\\\\\\ldots, n\\\\\\\\}$ appears precisely one time as a term of the sequence. For example, $(3, 5, 1, 2, 4)$ is a permutation of $\\\\\\\\{1, 2, 3, 4, 5\\\\\\\\}$. Let $P(n)$ be the number of permutations of $\\\\\\\\{1, 2, \\\\\\\\ldots, n\\\\\\\\}$ for which $ka_k$ is a perfect square for all $1\\\\\\\\leq k\\\\\\\\leq n$. Find with proof the smallest $n$ such that $P(n)$ is a multiple of $2010$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $a$, $b$, $c$ be positive real numbers. Prove that\n",
      "\n",
      "$\\dfrac{(2a + b + c)^2}{2a^2 + (b + c)^2} + \\dfrac{(2b + c + a)^2}{2b^2 + (c + a)^2} + \\dfrac{(2c + a + b)^2}{2c^2 + (a + b)^2} \\le 8.$\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"What is the smallest positive integer that can be expressed as the sum of nine consecutive integers, the sum of ten consecutive integers, and the sum of eleven consecutive integers?\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Suppose that $(a_1,b_1),$ $(a_2,b_2),$ $\\\\\\\\dots,$ $(a_{100},b_{100})$ are distinct ordered pairs of nonnegative integers. Let $N$ denote the number of pairs of integers $(i,j)$ satisfying $1\\\\\\\\leq i<j\\\\\\\\leq 100$ and $|a_ib_j-a_jb_i|=1$. Determine the largest possible value of $N$ over all possible choices of the $100$ ordered pairs.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"If $a$ and $b$ are two of the roots of $x^4+x^3-1=0$, prove that $ab$ is a root of $x^6+x^4+x^3-x^2-1=0$.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Find, with proof, the number of positive integers whose base-$n$ representation consists of distinct digits with the property that, except for the leftmost digit, every digit differs by $\\\\\\\\pm 1$ from some digit further to the left.  (Your answer should be an explicit function of $n$ in simplest form.)\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"A + B + C is an integral multiple of \\\\\\\\pi. x, y, and z are real numbers. If x\\\\\\\\sin(A) + y\\\\\\\\sin(B) + z\\\\\\\\sin(C) = x^2\\\\\\\\sin(2A) + y^2\\\\\\\\sin(2B) + z^2\\\\\\\\sin(2C) = 0, show that x^n\\\\\\\\sin(nA) + y^n\\\\\\\\sin(nB) + z^n\\\\\\\\sin(nC) = 0 for any positive integer n.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "If $a$ and $b$ are two of the roots of $x^4+x^3-1=0$, prove that $ab$ is a root of $x^6+x^4+x^3-x^2-1=0$.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"For any nonempty set $S$ of numbers, let $\\\\\\\\sigma(S)$ and $\\\\\\\\pi(S)$ denote the sum and product, respectively, of the elements of $S$. Prove that\\\\n\\\\\\\\[\\\\\\\\sum \\\\\\\\frac{\\\\\\\\sigma(S)}{\\\\\\\\pi(S)} = (n^2 + 2n) - \\\\\\\\left( 1 + \\\\\\\\frac{1}{2} + \\\\\\\\frac{1}{3} + \\\\\\\\cdots + \\\\\\\\frac{1}{n} \\\\\\\\right)  (n+1),\\\\\\\\]where \\\\\"$\\\\\\\\Sigma$\\\\\" denotes a sum involving all nonempty subsets $S$ of $\\\\\\\\{1,2,3, \\\\\\\\ldots,n\\\\\\\\}$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "$A + B + C$ is an integral multiple of $\\pi$. $x, y,$ and $z$ are real numbers. If $x\\sin(A)+y\\sin(B)+z\\sin(C)=x^2\\sin(2A)+y^2\\sin(2B)+z^2\\sin(2C)=0$, show that $x^n\\sin(nA)+y^n \\sin(nB) +z^n \\sin(nC)=0$ for any positive integer $n$.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Prove that there exists a positive integer $n < 10^6$ such that $5^n$ has six consecutive zeros in its decimal representation.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Suppose that necklace $A$ has 14 beads and necklace $B$ has 19. Prove that for any odd integer $n \\\\\\\\geq 1$, there is a way to number each of the 33 beads with an integer from the sequence \\\\\\\\{ n, n+1, n+2, \\\\\\\\dots, n+32 \\\\\\\\} so that each integer is used once, and adjacent beads correspond to relatively prime integers. (Here a \\\\\"necklace\\\\\" is viewed as a circle in which each bead is adjacent to two other beads.)\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Two permutations $a_1, a_2, \\\\\\\\ldots, a_{2010}$ and\\\\n$b_1, b_2, \\\\\\\\ldots, b_{2010}$ of the numbers $1, 2, \\\\\\\\ldots, 2010$\\\\nare said to intersect if $a_k = b_k$ for some value of $k$ in the\\\\nrange $1 \\\\\\\\le k\\\\\\\\le 2010$.  Show that there exist $1006$ permutations\\\\nof the numbers $1, 2, \\\\\\\\ldots, 2010$ such that any other such\\\\npermutation is guaranteed to intersect at least one of these $1006$\\\\npermutations.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Let \\\\\\\\(P\\\\\\\\) be a point in the plane of triangle \\\\\\\\(ABC\\\\\\\\), and \\\\\\\\(\\\\\\\\gamma\\\\\\\\) a line passing through \\\\\\\\(P\\\\\\\\).  Let \\\\\\\\(A\\'\\\\\\\\), \\\\\\\\(B\\'\\\\\\\\), \\\\\\\\(C\\'\\\\\\\\) be the points where the reflections of lines \\\\\\\\(PA\\\\\\\\), \\\\\\\\(PB\\\\\\\\), \\\\\\\\(PC\\\\\\\\) with respect to \\\\\\\\(\\\\\\\\gamma\\\\\\\\) intersect lines \\\\\\\\(BC\\\\\\\\), \\\\\\\\(AC\\\\\\\\), \\\\\\\\(AB\\\\\\\\), respectively.  Prove that \\\\\\\\(A\\'\\\\\\\\), \\\\\\\\(B\\'\\\\\\\\), \\\\\\\\(C\\'\\\\\\\\) are collinear.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $P$ be a point in the plane of triangle $ABC$, and $\\gamma$ a line passing through $P$.  Let $A'$, $B'$, $C'$ be the points where the reflections of lines $PA$, $PB$, $PC$ with respect to $\\gamma$ intersect lines $BC$, $AC$, $AB$, respectively.  Prove that $A'$, $B'$, $C'$ are collinear.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Which of the five \\\\\"T-like shapes\\\\\" would be symmetric to the one shown with respect to the dashed line?\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Prove \\\\\\\\[\\\\n\\\\\\\\frac{1}{\\\\\\\\cos 0^\\\\\\\\circ \\\\\\\\cos 1^\\\\\\\\circ} + \\\\\\\\frac{1}{\\\\\\\\cos 1^\\\\\\\\circ \\\\\\\\cos 2^\\\\\\\\circ} + \\\\\\\\cdots + \\\\\\\\frac{1}{\\\\\\\\cos 88^\\\\\\\\circ \\\\\\\\cos 89^\\\\\\\\circ} = \\\\\\\\frac{\\\\\\\\cos 1^\\\\\\\\circ}{\\\\\\\\sin^2 1^\\\\\\\\circ}.\\\\n\\\\\\\\]\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Each of eight boxes contains six balls. Each ball has been colored with one of $n$ colors, such that no two balls in the same box are the same color, and no two colors occur together in more than one box. Determine, with justification, the smallest integer $n$ for which this is possible.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $ABCD$ be a convex quadrilateral such that diagonals $AC$ and $BD$ intersect at right angles, and let $E$ be their intersection. Prove that the reflections of $E$ across $AB$, $BC$, $CD$, $DA$ are concyclic.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Prove that if the opposite sides of a skew (non-planar) quadrilateral are congruent, then the line joining the midpoints of the two diagonals is perpendicular to these diagonals, and conversely, if the line joining the midpoints of the two diagonals of a skew quadrilateral is perpendicular to these diagonals, then the opposite sides of the quadrilateral are congruent.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter $AB$. Denote by $P, Q, R, S$ the feet of the perpendiculars from $Y$ onto lines $AX, BX, AZ, BZ$, respectively. Prove that the acute angle formed by lines $PQ$ and $RS$ is half the size of $\\\\\\\\angle XOZ$, where $O$ is the midpoint of segment $AB$.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"For $n \\\\\\\\ge 2$ let $a_1$, $a_2$, ..., $a_n$ be positive real numbers such that $(a_1+a_2+ ... +a_n)\\\\\\\\left( \\\\\\\\frac{1}{a_1} + \\\\\\\\frac{1}{a_2} + ... +\\\\\\\\frac{1}{a_n} \\\\\\\\right) \\\\\\\\le \\\\\\\\left(n+ \\\\\\\\frac{1}{2} \\\\\\\\right)^2$.\\\\nProve that $\\\\\\\\text{max}(a_1, a_2, ... ,a_n) \\\\\\\\le  4 \\\\\\\\text{min}(a_1, a_2, ... , a_n)$.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $ABCD$ be a convex quadrilateral such that diagonals $AC$ and $BD$ intersect at right angles, and let $E$ be their intersection. Prove that the reflections of $E$ across $AB$, $BC$, $CD$, $DA$ are concyclic.\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Prove that if the opposite sides of a skew (non-planar) quadrilateral are congruent, then the line joining the midpoints of the two diagonals is perpendicular to these diagonals, and conversely, if the line joining the midpoints of the two diagonals of a skew quadrilateral is perpendicular to these diagonals, then the opposite sides of the quadrilateral are congruent.\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter\n",
      "$AB$. Denote by $P, Q, R, S$ the feet of the perpendiculars from $Y$ onto\n",
      "lines $AX, BX, AZ, BZ$, respectively. Prove that the acute angle\n",
      "formed by lines $PQ$ and $RS$ is half the size of $\\angle XOZ$, where\n",
      "$O$ is the midpoint of segment $AB$.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Points $A$, $B$, $C$, $D$, $E$ lie on a circle $\\\\\\\\omega$ and point $P$ lies outside the circle. The given points are such that (i) lines $PB$ and $PD$ are tangent to $\\\\\\\\omega$, (ii) $P$, $A$, $C$ are collinear, and (iii) $\\\\\\\\overline{DE} \\\\\\\\parallel \\\\\\\\overline{AC}$. Prove that $\\\\\\\\overline{BE}$ bisects $\\\\\\\\overline{AC}$.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Two boundary points of a ball of radius 1 are joined by a curve contained in the ball and having length less than 2. Prove that the curve is contained entirely within some hemisphere of the given ball.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $ABC$ be a triangle, and $M$ an interior point such that $\\\\\\\\angle MAB=10^\\\\\\\\circ$, $\\\\\\\\angle MBA=20^\\\\\\\\circ$ , $\\\\\\\\angle MAC= 40^\\\\\\\\circ$ and $\\\\\\\\angle MCA=30^\\\\\\\\circ$. Prove that the triangle is isosceles.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Let $p$ be a prime, and let $a_1, \\\\\\\\dots, a_p$ be integers. Show that there exists an integer $k$ such that the numbers $a_1 + k, a_2 + 2k, \\\\\\\\dots, a_p + pk$ produce at least $\\\\\\\\frac{1}{2} p$ distinct remainders upon division by $p$.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Points $A$, $B$, $C$, $D$, $E$ lie on a circle $\\omega$ and point $P$ lies outside the circle.  The given points are such that (i) lines $PB$ and $PD$ are tangent to $\\omega$, (ii) $P$, $A$, $C$ are collinear, and (iii) $\\overline{DE} \\parallel \\overline{AC}$.  Prove that $\\overline{BE}$ bisects $\\overline{AC}$.\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Two boundary points of a ball of radius 1 are joined by a curve contained in the ball and having length less than 2. Prove that the curve is contained entirely within some hemisphere of the given ball.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Show that for any triangle, $\\\\\\\\frac{3\\\\\\\\sqrt{3}}{2} \\\\\\\\ge \\\\\\\\sin(3A) + \\\\\\\\sin(3B) + \\\\\\\\sin(3C) \\\\\\\\ge -2$. \\\\nWhen does the equality hold?\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $a, b$ be integers greater than 2.  Prove that there exists a positive integer $k$ and a finite sequence $n_1, n_2, \\\\\\\\ldots, n_k$ of positive integers such that $n_1 = a$, $n_k = b$, and $n_in_{i+1}$ is divisible by $n_i + n_{i+1}$ for each $i$ ($1 \\\\\\\\le i \\\\\\\\le k$).\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $ABC$ be a triangle, and $M$ an interior point such that $\\angle MAB=10^\\circ$, $\\angle MBA=20^\\circ$ , $\\angle MAC= 40^\\circ$ and $\\angle MCA=30^\\circ$. Prove that the triangle is isosceles.\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $p$ be a prime, and let $a_1, \\dots, a_p$ be integers. Show that there exists an integer $k$ such that the numbers \\[a_1 + k, a_2 + 2k, \\dots, a_p + pk\\]produce at least $\\tfrac{1}{2} p$ distinct remainders upon division by $p$.\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "All of the triangles in the diagram below are similar to isosceles triangle $ABC$, in which $AB=AC$. Each of the $7$ smallest triangles has area $1,$ and $\\triangle ABC$ has area $40$. What is the area of trapezoid $DBCE$?\n",
      "\n",
      "$\\textbf{(A) }   16   \\qquad        \\textbf{(B) }   18   \\qquad    \\textbf{(C) }   20   \\qquad   \\textbf{(D) }  22 \\qquad  \\textbf{(E) }   24$\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $n \\\\\\\\geq 2$ be an integer. Carl has $n$ books arranged on a bookshelf.\\\\nEach book has a height and a width. No two books have the same height, and no two\\\\nbooks have the same width.\\\\nInitially, the books are arranged in increasing order of height from left to right. In a\\\\nmove, Carl picks any two adjacent books where the left book is wider and shorter than\\\\nthe right book, and swaps their locations. Carl does this repeatedly until no further\\\\nmoves are possible.\\\\nProve that regardless of how Carl makes his moves, he must stop after a finite number\\\\nof moves, and when he does stop, the books are sorted in increasing order of width\\\\nfrom left to right.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $a, b$ be integers greater than 2.  Prove that there exists a positive integer $k$ and a finite sequence $n_1, n_2, \\ldots, n_k$ of positive integers such that $n_1 = a$, $n_k = b$, and $n_in_{i+1}$ is divisible by $n_i + n_{i+1}$ for each $i$ ($1 \\le i \\le k$).\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Show that for any triangle, $\\frac{3\\sqrt{3}}{2}\\ge \\sin(3A) + \\sin(3B) + \\sin (3C) \\ge -2$. \n",
      "When does the equality hold?\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $n \\geq 2$ be an integer. Carl has $n$ books arranged on a bookshelf.\n",
      "Each book has a height and a width. No two books have the same height, and no two\n",
      "books have the same width.\n",
      "Initially, the books are arranged in increasing order of height from left to right. In a\n",
      "move, Carl picks any two adjacent books where the left book is wider and shorter than\n",
      "the right book, and swaps their locations. Carl does this repeatedly until no further\n",
      "moves are possible.\n",
      "Prove that regardless of how Carl makes his moves, he must stop after a finite number\n",
      "of moves, and when he does stop, the books are sorted in increasing order of width\n",
      "from left to right.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $\\\\\\\\mathbb{Z}$ be the set of integers. Find all functions $f : \\\\\\\\mathbb{Z} \\\\\\\\rightarrow \\\\\\\\mathbb{Z}$ such that\\\\n\\\\\\\\[xf(2f(y)-x)+y^2f(2x-f(y))=\\\\\\\\frac{f(x)^2}{x}+f(yf(y))\\\\\\\\] for all $x, y \\\\\\\\in \\\\\\\\mathbb{Z}$ with $x \\\\\\\\neq 0.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Let $ABC$ be a triangle with orthocenter $H$ and let $P$ be the second intersection of the circumcircle of triangle $AHC$ with the internal bisector of the angle $\\\\\\\\angle BAC$. Let $X$ be the circumcenter of triangle $APB$ and $Y$ the orthocenter of triangle $APC$. Prove that the length of segment $XY$ is equal to the circumradius of triangle $ABC$.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "How many 4-digit positive integers have four different digits, where the leading digit is not zero, the integer is a multiple of 5, and 5 is the largest digit?\n",
      "$\\textbf{(A) }24\\qquad\\textbf{(B) }48\\qquad\\textbf{(C) }60\\qquad\\textbf{(D) }84\\qquad\\textbf{(E) }108$\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "FAIL PARSINGThe symbols $(a,b,\\ldots,g)$ and $[a,b,\\ldots, g]$ denote the greatest common divisor and least common multiple, respectively, of the positive integers $a,b,\\ldots, g$. For example, $(3,6,18)=3$ and $[6,15]=30$. Prove that\n",
      "\n",
      "$\\frac{[a,b,c]^2}{[a,b][b,c][c,a]}=\\frac{(a,b,c)^2}{(a,b)(b,c)(c,a)}$.\n",
      "\n",
      "'```json\\n{\\n\"problem\": \"Let $n \\\\\\\\neq 0$. For every sequence of integers\\\\n\\\\n$A = a_0,a_1,a_2,\\\\\\\\dots, a_n$\\\\n\\\\nsatisfying $0 \\\\\\\\le a_i \\\\\\\\le i$, for $i=0,\\\\\\\\dots,n$, define another sequence\\\\n\\\\n$t(A)= t(a_0), t(a_1), t(a_2), \\\\\\\\dots, t(a_n)$\\\\n\\\\nby setting $t(a_i)$ to be the number of  terms in the sequence $A$ that precede the term $a_i$ and are different from $a_i$. Show that, starting from any sequence $A$ as above, fewer than $n$ applications of the transformation $t$ lead to a sequence $B$ such that $t(B) = B$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"If this path is to continue in the same pattern, then which sequence of arrows goes from point $425$ to point $427$?\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"A list of $8$ numbers is formed by beginning with two given numbers.  Each new number in the list is the product of the two previous numbers.  Find the first number if the last three are shown:\\\\n\\\\[\\\\text{\\\\underline{\\\\hspace{3 mm}?\\\\hspace{3 mm}}\\\\hspace{1 mm},\\\\hspace{1 mm} \\\\underline{\\\\hspace{7 mm}}\\\\hspace{1 mm},\\\\hspace{1 mm} \\\\underline{\\\\hspace{7 mm}}\\\\hspace{1 mm},\\\\hspace{1 mm} \\\\underline{\\\\hspace{7 mm}}\\\\hspace{1 mm},\\\\hspace{1 mm} \\\\underline{\\\\hspace{7 mm}}\\\\hspace{1 mm},\\\\hspace{1 mm}\\\\underline{\\\\hspace{2 mm}16\\\\hspace{2 mm}}\\\\hspace{1 mm},\\\\hspace{1 mm}\\\\underline{\\\\hspace{2 mm}64\\\\hspace{2 mm}}\\\\hspace{1 mm},\\\\hspace{1 mm}\\\\underline{\\\\hspace{1 mm}1024\\\\hspace{1 mm}}}\\\\]\",\\n\"A\": \"\\\\\\\\frac{1}{64}\",\\n\"B\": \"\\\\\\\\frac{1}{4}\",\\n\"C\": \"1\",\\n\"D\": \"2\",\\n\"E\": \"4\"\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Suppose that the set $\\\\\\\\{1,2,\\\\\\\\cdots, 1998\\\\\\\\}$ has been partitioned into disjoint pairs $\\\\\\\\{a_i,b_i\\\\\\\\}$ ($1\\\\\\\\leq i\\\\\\\\leq 999$) so that for all $i$, $|a_i-b_i|$ equals $1$ or $6$. Prove that the sum $|a_1-b_1|+|a_2-b_2|+\\\\\\\\cdots +|a_{999}-b_{999}|$ ends in the digit $9$.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $n \\neq 0$. For every sequence of integers\n",
      "\n",
      "\n",
      "$A = a_0,a_1,a_2,\\dots, a_n$\n",
      "\n",
      "\n",
      "satisfying $0 \\le a_i \\le i$, for $i=0,\\dots,n$, define another sequence\n",
      "\n",
      "\n",
      "$t(A)= t(a_0), t(a_1), t(a_2), \\dots, t(a_n)$\n",
      "\n",
      "\n",
      "by setting $t(a_i)$ to be the number of  terms in the sequence $A$ that precede the term $a_i$ and are different from $a_i$. Show that, starting from any sequence $A$ as above, fewer than $n$ applications of the transformation $t$ lead to a sequence $B$ such that $t(B) = B$.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Consider the assertion that for each positive integer $n \\\\\\\\ge 2$, the remainder upon dividing $2^{2^n}$ by $2^n-1$ is a power of 4. Either prove the assertion or find (with proof) a counter-example.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Suppose that the set $\\{1,2,\\cdots, 1998\\}$ has been partitioned into disjoint pairs $\\{a_i,b_i\\}$ ($1\\leq i\\leq 999$) so that for all $i$, $|a_i-b_i|$ equals $1$ or $6$. Prove that the sum \\[|a_1-b_1|+|a_2-b_2|+\\cdots +|a_{999}-b_{999}|\\] ends in the digit $9$.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"For a given integer $n \\\\\\\\ge 2,$ let $\\\\\\\\{a_1,a_2,\\\\\\\\dots,a_m\\\\\\\\}$ be the set of positive integers less than $n$ that are relatively prime to $n.$ Prove that if every prime that divides $m$ also divides $n,$ then $a_1^k+a_2^k + \\\\\\\\dots + a_m^k$ is divisible by $m$ for every positive integer $k.$\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "For a given integer $n\\ge 2,$ let $\\{a_1,a_2,…,a_m\\}$ be the set of positive integers less than $n$ that are relatively prime to $n.$ Prove that if every prime that divides $m$ also divides $n,$ then $a_1^k+a_2^k + \\dots + a_m^k$ is divisible by $m$ for every positive integer $k.$\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"An acute-angled triangle $ABC$ is given in the plane. The circle with diameter $AB$ intersects altitude $CC\\'$ and its extension at points $M$ and $N$, and the circle with diameter $AC$ intersects altitude $BB\\'$ and its extensions at $P$ and $Q$. Prove that the points $M, N, P, Q$ lie on a common circle.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"(Kiran Kedlaya) Three nonnegative real numbers $r_1$, $r_2$, $r_3$ are written on a blackboard. These numbers have the property that there exist integers $a_1$, $a_2$, $a_3$, not all zero, satisfying $a_1r_1 + a_2r_2 + a_3r_3 = 0$. We are permitted to perform the following operation: find two numbers $x$, $y$ on the blackboard with $x \\\\\\\\le y$, then erase $y$ and write $y - x$ in its place. Prove that after a finite number of such operations, we can end up with at least one $0$ on the blackboard.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Let $n \\\\\\\\geq 5$ be an integer. Find the largest integer $k$ (as a function of $n$) such that there exists a convex $n$-gon $A_{1}A_{2}\\\\\\\\dots A_{n}$ for which exactly $k$ of the quadrilaterals $A_{i}A_{i+1}A_{i+2}A_{i+3}$ have an inscribed circle. (Here $A_{n+j} = A_{j}$.)\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "An acute-angled triangle $ABC$ is given in the plane. The circle with diameter $\\, AB \\,$ intersects altitude $\\, CC' \\,$ and its extension at points $\\, M \\,$ and $\\, N \\,$, and the circle with diameter $\\, AC \\,$ intersects altitude $\\, BB' \\,$ and its extensions at $\\, P \\,$ and $\\, Q \\,$. Prove that the points $\\, M, N, P, Q \\,$ lie on a common circle.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"A circle $\\\\\\\\omega$ is inscribed in a quadrilateral $ABCD$.  Let $I$ be the center of $\\\\\\\\omega$.  Suppose that\\\\n\\\\n$(AI + DI)^2 + (BI + CI)^2 = (AB + CD)^2$.\\\\n\\\\nProve that $ABCD$ is an isosceles trapezoid.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Suppose that $(a_1, b_1), (a_2, b_2), \\\\\\\\ldots , (a_{100}, b_{100})$ are distinct ordered pairs of nonnegative integers. Let $N$ denote the number of pairs of integers $(i, j)$ satisfying $1 \\\\\\\\le i < j \\\\\\\\le 100$ and $|a_ib_j - a_j b_i|=1$. Determine the largest possible value of $N$ over all possible choices of the $100$ ordered pairs.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"(Dick Gibbs) For a given positive integer $k$ find, in terms of $k$, the minimum value of $N$ for which there is a set of $2k+1$ distinct positive integers that has sum greater than $N$ but every subset of size $k$ has sum at most $\\\\\\\\frac{N}{2}$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Let $P$ be a point in the plane of triangle $ABC$, and $\\\\\\\\gamma$ a line passing through $P$. Let $A\\'$, $B\\'$, $C\\'$ be the points where the reflections of lines $PA$, $PB$, $PC$ with respect to $\\\\\\\\gamma$ intersect lines $BC$, $AC$, $AB$, respectively. Prove that $A\\'$, $B\\'$, $C\\'$ are collinear.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"The sides of a $99$-gon are initially colored so that consecutive sides are red, blue, red, blue,..., red, blue, yellow. We make a sequence of modifications in the coloring, changing the color of one side at a time to one of the three given colors (red, blue, yellow), under the constraint that no two adjacent sides may be the same color. By making a sequence of such modifications, is it possible to arrive at the coloring in which consecutive sides are red, blue, red, blue, red, blue,..., red, yellow, blue?\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Let $X_1, X_2, \\\\\\\\ldots, X_{100}$ be a sequence of mutually distinct nonempty subsets of a set $S$. Any two sets $X_i$ and $X_{i+1}$ are disjoint and their union is not the whole set $S$, that is, $X_i\\\\\\\\cap X_{i+1}=\\\\\\\\emptyset$ and $X_i\\\\\\\\cup X_{i+1}\\\\\\\\neq S$, for all $i\\\\\\\\in\\\\\\\\{1, \\\\\\\\ldots, 99\\\\\\\\}$. Find the smallest possible number of elements in $S$.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $P$ be a point in the plane of triangle $ABC$, and $\\gamma$ a line passing through $P$.  Let $A'$, $B'$, $C'$ be the points where the reflections of lines $PA$, $PB$, $PC$ with respect to $\\gamma$ intersect lines $BC$, $AC$, $AB$, respectively.  Prove that $A'$, $B'$, $C'$ are collinear.\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Find the sum of all prime numbers between $1$ and $100$ that are simultaneously $1$ greater than a multiple of $4$ and $1$ less than a multiple of $5$. \n",
      "$\\mathrm{(A) \\ } 118 \\qquad \\mathrm{(B) \\ }137 \\qquad \\mathrm{(C) \\ } 158 \\qquad \\mathrm{(D) \\ } 187 \\qquad \\mathrm{(E) \\ } 245$\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Una rolls $6$ standard $6$-sided dice simultaneously and calculates the product of the $6{ }$ numbers obtained. What is the probability that the product is divisible by $4?$\n",
      "$\\textbf{(A)}\\: \\frac34\\qquad\\textbf{(B)} \\: \\frac{57}{64}\\qquad\\textbf{(C)} \\: \\frac{59}{64}\\qquad\\textbf{(D)} \\: \\frac{187}{192}\\qquad\\textbf{(E)} \\: \\frac{63}{64}$\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Consider the equation \\\\n\\\\\\\\[\\\\\\\\left(3x^3 + xy^2 \\\\\\\\right) \\\\\\\\left(x^2y + 3y^3 \\\\\\\\right) = (x-y)^7.\\\\\\\\]\\\\n(a) Prove that there are infinitely many pairs $(x,y)$ of positive integers satisfying the equation. \\\\n(b) Describe all pairs $(x,y)$ of positive integers satisfying the equation.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Carina has three pins, labeled $A, B$, and $C$, respectively, located at the origin of the coordinate plane. In a move, Carina may move a pin to an adjacent lattice point at distance $1$ away. What is the least number of moves that Carina can make in order for triangle $ABC$ to have area 2021?\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"I have an $n \\\\\\\\times n$ sheet of stamps, from which I\\'ve been asked to tear out blocks of three adjacent stamps in a single row or column. (I can only tear along the perforations separating adjacent stamps, and each block must come out of the sheet in one piece.) Let $b(n)$ be the smallest number of blocks I can tear out and make it impossible to tear out any more blocks. Prove that there are real constants $c$ and $d$ such that $\\\\\\\\frac{1}{7} n^2 - cn \\\\\\\\leq b(n) \\\\\\\\leq \\\\\\\\frac{1}{5} n^2 + dn$ for all $n > 0$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Find all integers $n \\\\\\\\ge 3$ such that among any $n$ positive real numbers $a_1$, $a_2$, $\\\\\\\\dots$, $a_n$ with\\\\n\\\\\\\\[\\\\\\\\max(a_1, a_2, \\\\\\\\dots, a_n) \\\\\\\\le n \\\\\\\\cdot \\\\\\\\min(a_1, a_2, \\\\\\\\dots, a_n),\\\\\\\\]there exist three that are the side lengths of an acute triangle.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Let $S_r = x^r + y^r + z^r$ with $x, y, z$ real. It is known that if $S_1 = 0$,\\\\n$(*)$ $\\\\\\\\frac{S_{m+n}}{m+n} = \\\\\\\\frac{S_m}{m} \\\\\\\\frac{S_n}{n}$\\\\nfor $(m, n) = (2, 3), (3, 2), (2, 5)$, or $(5, 2)$. Determine all other pairs of integers $(m, n)$ if any, so that $(*)$ holds for all real numbers $x, y, z$ such that $x + y + z = 0$.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"The nonzero coefficients of a polynomial $P$ with real coefficients are all replaced by their mean to form a polynomial $Q$. Which of the following could be a graph of $y = P(x)$ and $y = Q(x)$ over the interval $-4 \\\\\\\\leq x \\\\\\\\leq 4$?\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "Gemini not happy.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"For distinct positive integers $a$, $b < 2012$, define $f(a,b)$ to be the number of integers $k$ with $1 \\\\\\\\le k < 2012$ such that the remainder when $ak$ divided by 2012 is greater than that of $bk$ divided by 2012. Let $S$ be the minimum value of $f(a,b)$, where $a$ and $b$ range over all pairs of distinct positive integers less than 2012. Determine $S$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $S$ be the sum of all numbers of the form $a/b,$ where $a$ and $b$ are relatively prime positive divisors of $1000.$ What is the greatest integer that does not exceed $S/10$?\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"$X$ is the smallest set of polynomials $p(x)$ such that: \\\\n\\\\n1. $p(x) = x$ belongs to $X$.\\\\n2. If $r(x)$ belongs to $X$, then $x\\\\\\\\cdot r(x)$ and $(x + (1 - x) \\\\\\\\cdot r(x) )$ both belong to $X$.\\\\nShow that if $r(x)$ and $s(x)$ are distinct elements of $X$, then $r(x) \\\\\\\\neq s(x)$ for any $0 < x < 1$.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"The Y2K Game is played on a $1 \\\\\\\\times 2000$ grid as follows. Two players in turn write either an S or an O in an empty square. The first player who produces three consecutive boxes that spell SOS wins. If all boxes are filled without producing SOS then the game is a draw. Prove that the second player has a winning strategy.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Show that the cube roots of three distinct prime numbers cannot be three terms (not necessarily consecutive) of an arithmetic progression.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "$X$ is the smallest set of polynomials $p(x)$ such that: \n",
      "\n",
      "1. $p(x) = x$ belongs to $X$.\n",
      "2. If $r(x)$ belongs to $X$, then $x\\cdot r(x)$ and $(x + (1 - x) \\cdot r(x) )$ both belong to $X$.\n",
      "Show that if $r(x)$ and $s(x)$ are distinct elements of $X$, then $r(x) \\neq s(x)$ for any $0 < x < 1$.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $OABC$ be a unit square in the $xy$-plane with $O(0,0), A(1,0), B(1,1)$ and $C(0,1)$. Let $u = x^2 - y^2$, and $v = xy$ be a transformation of the $xy$-plane into the $uv$-plane. The transform (or image) of the square is:\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "The Y2K Game is played on a $1 \\times 2000$ grid as follows. Two players in turn write either an S or an O in an empty square. The first player who produces three consecutive boxes that spell SOS wins. If all boxes are filled without producing SOS then the game is a draw. Prove that the second player has a winning strategy.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Let $k_1 < k_2 < k_3 < \\\\\\\\cdots$ be positive integers, no two consecutive, and let $s_m = k_1 + k_2 + \\\\\\\\cdots + k_m$ for $m = 1, 2, 3, \\\\\\\\ldots$. Prove that, for each positive integer $n$, the interval $[s_n, s_{n+1})$ contains at least one perfect square.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Find all functions $f : \\\\\\\\mathbb{Z}^+ \\\\\\\\to \\\\\\\\mathbb{Z}^+$ (where $\\\\\\\\mathbb{Z}^+$ is the set of positive integers) such that $f(n!) = f(n)!$ for all positive integers $n$ and such that $m - n$ divides $f(m) - f(n)$ for all distinct positive integers $m$, $n$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Suppose that in a certain society, each pair of persons can be classified as either amicable or hostile. We shall say that each member of an amicable pair is a friend of the other, and each member of a hostile pair is a foe of the other. Suppose that the society has $n$ persons and $q$ amicable pairs, and that for every set of three persons, at least one pair is hostile. Prove that there is at least one member of the society whose foes include $q(1 - 4q/n^2)$ or fewer amicable pairs.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Find the sum of all positive rational numbers that are less than 10 and that have denominator 30 when written in lowest terms.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $A$ be a set with $|A| = 225$, meaning that $A$ has 225 elements. Suppose further that there are eleven subsets $A_1$, ..., $A_{11}$ of $A$ such that $|A_i| = 45$ for $1 \\\\\\\\le i \\\\\\\\le 11$ and $|A_i \\\\\\\\cap A_j| = 9$ for $1 \\\\\\\\le i < j \\\\\\\\le 11$. Prove that $|A_1 \\\\\\\\cup A_2 \\\\\\\\cup ... \\\\\\\\cup A_{11}| \\\\\\\\ge 165$, and give an example for which equality holds.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $ABC$ be a triangle with incenter $I$, incircle $\\\\\\\\gamma$ and circumcircle $\\\\\\\\Gamma$. Let $M, N, P$ be the midpoints of sides $\\\\\\\\overline{BC}$, $\\\\\\\\overline{CA}$, $\\\\\\\\overline{AB}$ and let $E, F$ be the tangency points of $\\\\\\\\gamma$ with $\\\\\\\\overline{CA}$ and $\\\\\\\\overline{AB}$, respectively. Let $U, V$ be the intersections of line $EF$ with line $MN$ and line $MP$, respectively, and let $X$ be the midpoint of arc $BAC$ of $\\\\\\\\Gamma$.\\\\n(a) Prove that $I$ lies on ray $CV$.\\\\n(b) Prove that line $XI$ bisects $\\\\\\\\overline{UV}$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $\\, k_1 < k_2 < k_3 <\\cdots\\,$, be positive integers, no two consecutive, and let $\\, s_m = k_1+k_2+\\cdots+k_m\\,$, for $\\, m = 1,2,3,\\ldots\\;\\;$. Prove that, for each positive integer $n$, the interval $\\, [s_n, s_{n+1})\\,$, contains at least one perfect square.\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Under the new AMC $10, 12$ scoring method, $6$ points are given for each correct answer, $2.5$ points are given for each unanswered question, and no points are given for an incorrect answer. Some of the possible scores between $0$ and $150$ can be obtained in only one way, for example, a score of $104.5$ can be obtained with $17$ correct answers, $1$ unanswered question, and $7$ incorrect answers, and also with $12$ correct answers and $13$ unanswered questions. There are scores that can be obtained in exactly three ways. What is their sum?\n",
      "$\\text{(A) }175 \\qquad \\text{(B) }179.5 \\qquad \\text{(C) }182 \\qquad \\text{(D) }188.5 \\qquad \\text{(E) }201$\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"(Titu Andreescu) Prove that for each positive integer $n$, there are pairwise relatively prime integers $k_0, k_1, \\\\\\\\dots, k_n$, all strictly greater than 1, such that $k_0 k_1 \\\\\\\\dotsm k_n -1$ is the product of two consecutive integers.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"An equilateral pentagon $AMNPQ$ is inscribed in triangle $ABC$ such that $M \\\\\\\\in \\\\\\\\overline{AB},$ $Q \\\\\\\\in \\\\\\\\overline{AC},$ and $N, P \\\\\\\\in \\\\\\\\overline{BC}.$ Let $S$ be the intersection of lines $MN$ and $PQ.$ Denote by $\\\\\\\\ell$ the angle bisector of $\\\\\\\\angle MSQ.$ Prove that $\\\\\\\\overline{OI}$ is parallel to $\\\\\\\\ell,$ where $O$ is the circumcenter of triangle $ABC,$ and $I$ is the incenter of triangle $ABC.$\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "FAIL PARSING'```json\\n{\\n \"problem\": \"How many positive integers less than $1000$ are $6$ times the sum of their digits?\",\\n \"A\": \"0\",\\n \"B\": \"1\",\\n \"C\": \"2\",\\n \"D\": \"4\",\\n \"E\": null\\n}\\n```'\n",
      "\n",
      "'```json\\n{\\n\"problem\": \"Steve is piling $m\\\\\\\\geq 1$ indistinguishable stones on the squares of an $n\\\\\\\\times n$ grid. Each square can have an arbitrarily high pile of stones. After he finished piling his stones in some manner, he can then perform stone moves, defined as follows. Consider any four grid squares, which are corners of a rectangle, i.e. in positions $(i, k), (i, l), (j, k), (j, l)$ for some $1\\\\\\\\leq i, j, k, l\\\\\\\\leq n$, such that $i<j$ and $k<l$. A stone move consists of either removing one stone from each of $(i, k)$ and $(j, l)$ and moving them to $(i, l)$ and $(j, k)$ respectively, or removing one stone from each of $(i, l)$ and $(j, k)$ and moving them to $(i, k)$ and $(j, l)$ respectively.\\\\nTwo ways of piling the stones are equivalent if they can be obtained from one another by a sequence of stone moves.\\\\nHow many different non-equivalent ways can Steve pile the stones on the grid?\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "(Titu Andreescu) Prove that for each positive integer $n$, there are pairwise relatively prime integers $k_0, k_1 \\dotsc, k_n$, all strictly greater than 1, such that $k_0 k_1 \\dotsm k_n -1$ is the product of two consecutive integers.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"In convex cyclic quadrilateral $ABCD,$ we know that lines $AC$ and $BD$ intersect at $E,$ lines $AB$ and $CD$ intersect at $F,$ and lines $BC$ and $DA$ intersect at $G.$ Suppose that the circumcircle of $\\\\\\\\triangle ABE$ intersects line $CB$ at $B$ and $P$, and the circumcircle of $\\\\\\\\triangle ADE$ intersects line $CD$ at $D$ and $Q$, where $C,B,P,G$ and $C,Q,D,F$ are collinear in that order. Prove that if lines $FP$ and $GQ$ intersect at $M$, then $\\\\\\\\angle MAC = 90^{\\\\\\\\circ}.$\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Prove that for any integer $n$, there exists a unique polynomial $Q(x)$ with coefficients in $\\\\\\\\{0,1,...,9\\\\\\\\}$ such that $Q(-2)=Q(-5)=n$.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Consider the two triangles $\\\\\\\\triangle ABC$ and $\\\\\\\\triangle PQR$ shown in Figure 1. In $\\\\\\\\triangle ABC$, $\\\\\\\\angle ADB = \\\\\\\\angle BDC = \\\\\\\\angle CDA = 120^\\\\\\\\circ$. Prove that $x = u + v + w$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "An equilateral pentagon $AMNPQ$ is inscribed in triangle $ABC$ such that $M\\in\\overline{AB},$ $Q\\in\\overline{AC},$ and $N, P\\in\\overline{BC}.$ Let $S$ be the intersection of lines $MN$ and $PQ.$ Denote by $\\ell$ the angle bisector of $\\angle MSQ.$\n",
      "Prove that $\\overline{OI}$ is parallel to $\\ell,$ where $O$ is the circumcenter of triangle $ABC,$ and $I$ is the incenter of triangle $ABC.$\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "In convex cyclic quadrilateral $ABCD,$ we know that lines $AC$ and $BD$ intersect at $E,$ lines $AB$ and $CD$ intersect at $F,$ and lines $BC$ and $DA$ intersect at $G.$ Suppose that the circumcircle of $\\triangle ABE$ intersects line $CB$ at $B$ and $P$, and the circumcircle of $\\triangle ADE$ intersects line $CD$ at $D$ and $Q$, where $C,B,P,G$ and $C,Q,D,F$ are collinear in that order. Prove that if lines $FP$ and $GQ$ intersect at $M$, then $\\angle MAC = 90^{\\circ}.$\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Consider the two triangles $\\triangle ABC$ and $\\triangle PQR$ shown in Figure 1.  In $\\triangle ABC$, $\\angle ADB = \\angle BDC = \\angle CDA = 120^\\circ$.  Prove that $x=u+v+w$.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Which cylinder has twice the volume of the cylinder shown above?\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": \"None of the above\"\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $n \\\\\\\\geq 4$ be an integer. Find all positive real solutions to the following system of $2n$ equations:\\\\n\\\\\\\\begin{align*} a_{1} &= \\\\\\\\frac{1}{a_{2 n}} + \\\\\\\\frac{1}{a_{2}}, & a_{2} &= a_{1} + a_{3}, \\\\\\\\\\\\\\\\ a_{3} &= \\\\\\\\frac{1}{a_{2}} + \\\\\\\\frac{1}{a_{4}}, & a_{4} &= a_{3} + a_{5}, \\\\\\\\\\\\\\\\ a_{5} &= \\\\\\\\frac{1}{a_{4}} + \\\\\\\\frac{1}{a_{6}}, & a_{6} &= a_{5} + a_{7}, \\\\\\\\\\\\\\\\ &\\\\\\\\vdots \\\\\\\\\\\\\\\\ a_{2 n-1} &= \\\\\\\\frac{1}{a_{2 n-2}} + \\\\\\\\frac{1}{a_{2 n}}, & a_{2 n} &= a_{2 n-1} + a_{1} \\\\\\\\end{align*}\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "The set $G$ is defined by the points $(x,y)$ with integer coordinates, $3\\le|x|\\le7$, $3\\le|y|\\le7$. How many squares of side at least $6$ have their four vertices in $G$?\n",
      "\n",
      "$\\textbf{(A)}\\ 125\\qquad \\textbf{(B)}\\ 150\\qquad \\textbf{(C)}\\ 175\\qquad \\textbf{(D)}\\ 200\\qquad \\textbf{(E)}\\ 225$\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $p$ be a prime, and let $a_1, \\\\\\\\dots, a_p$ be integers. Show that there exists an integer $k$ such that the numbers $a_1 + k, a_2 + 2k, \\\\\\\\dots, a_p + pk$ produce at least $\\\\\\\\frac{1}{2} p$ distinct remainders upon division by $p$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"A random number selector can only select one of the nine integers 1, 2, ..., 9, and it makes these selections with equal probability. Determine the probability that after $n$ selections ($n>1$), the product of the $n$ numbers selected will be divisible by 10.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $n$ be a nonnegative integer. Determine the number of ways that one can choose $(n+1)^2$ sets $S_{i,j} \\\\\\\\subseteq \\\\\\\\{1,2,\\\\\\\\ldots,2n\\\\\\\\}$, for integers $i,j$ with $0 \\\\\\\\leq i,j \\\\\\\\leq n$, such that:\\\\n$\\\\\\\\bullet$ for all $0 \\\\\\\\leq i,j \\\\\\\\leq n$, the set $S_{i,j}$ has $i+j$ elements; and\\\\n$\\\\\\\\bullet$ $S_{i,j} \\\\\\\\subseteq S_{k,l}$ whenever $0 \\\\\\\\leq i \\\\\\\\leq k \\\\\\\\leq n$ and $0 \\\\\\\\leq j \\\\\\\\leq l \\\\\\\\leq n$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $p$ be a prime, and let $a_1, \\dots, a_p$ be integers. Show that there exists an integer $k$ such that the numbers \\[a_1 + k, a_2 + 2k, \\dots, a_p + pk\\]produce at least $\\tfrac{1}{2} p$ distinct remainders upon division by $p$.\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Jerry starts at $0$ on the real number line. He tosses a fair coin $8$ times. When he gets heads, he moves $1$ unit in the positive direction; when he gets tails, he moves $1$ unit in the negative direction. The probability that he reaches $4$ at some time during this process $\\frac{a}{b},$ where $a$ and $b$ are relatively prime positive integers. What is $a + b?$ (For example, he succeeds if his sequence of tosses is $HTHHHHHH.$)\n",
      "$\\textbf{(A)}\\ 69\\qquad\\textbf{(B)}\\ 151\\qquad\\textbf{(C)}\\ 257\\qquad\\textbf{(D)}\\ 293\\qquad\\textbf{(E)}\\ 313$\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Determine which integers $n > 1$ have the property that there exists an infinite sequence $a_1$, $a_2$, $a_3$, $\\\\\\\\dots$ of nonzero integers such that the equality\\\\n\\\\\\\\[a_k + 2a_{2k} + \\\\\\\\dots + na_{nk} = 0\\\\\\\\]\\\\nholds for every positive integer $k$.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $a$, $b$ be odd positive integers. Define the sequence $(f_n)$ by putting $f_1 = a$,\\\\n$f_2 = b$, and by letting $f_n$ for $n\\\\\\\\ge3$ be the greatest odd divisor of $f_{n-1} + f_{n-2}$.\\\\nShow that $f_n$ is constant for $n$ sufficiently large and determine the eventual\\\\nvalue as a function of $a$ and $b$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Rectangles $BCC_1B_2,$ $CAA_1C_2,$ and $ABB_1A_2$ are erected outside an acute triangle $ABC.$ Suppose that $\\\\\\\\angle BC_1C+\\\\\\\\angle CA_1A+\\\\\\\\angle AB_1B=180^{\\\\\\\\circ}.$ Prove that lines $B_1C_2,$ $C_1A_2,$ and $A_1B_2$ are concurrent.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"If $a, b, c, d, e$ are positive numbers bounded by $p$ and $q$, i.e, if they lie in $[p,q], 0 < p$, prove that\\\\n\\\\n$\\\\\\\\quad (a+b +c +d +e)\\\\\\\\left(\\\\\\\\frac{1}{a} +\\\\\\\\frac {1}{b} +\\\\\\\\frac{1}{c} + \\\\\\\\frac{1}{d} +\\\\\\\\frac{1}{e}\\\\\\\\right) \\\\\\\\le 25 + 6\\\\\\\\left(\\\\\\\\sqrt{\\\\\\\\frac {p}{q}} - \\\\\\\\sqrt {\\\\\\\\frac{q}{p}}\\\\\\\\right)^2$\\\\n\\\\nand determine when there is equality.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Find the smallest positive integer $n$ such that if $n$ squares of a $1000 \\\\\\\\times 1000$ chessboard are colored, then there will exist three colored squares whose centers form a right triangle with sides parallel to the edges of the board.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Rectangles $BCC_1B_2,$ $CAA_1C_2,$ and $ABB_1A_2$ are erected outside an acute triangle $ABC.$ Suppose that\\[\\angle BC_1C+\\angle CA_1A+\\angle AB_1B=180^{\\circ}.\\]Prove that lines $B_1C_2,$ $C_1A_2,$ and $A_1B_2$ are concurrent.\n",
      "Retrying answer fetching:\n",
      "A\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "A pyramid has a square base with sides of length $1$ and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?\n",
      "$\\textbf{(A)}\\ 5\\sqrt{2} - 7 \\qquad\\textbf{(B)}\\ 7 - 4\\sqrt{3} \\qquad\\textbf{(C)}\\ \\frac{2\\sqrt{2}}{27} \\qquad\\textbf{(D)}\\ \\frac{\\sqrt{2}}{9} \\qquad\\textbf{(E)}\\ \\frac{\\sqrt{3}}{9}$\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"(Kiran Kedlaya) Suppose $a_1, \\\\\\\\dots, a_n$ are integers whose greatest common divisor is 1. Let $S$ be a set of integers with the following properties:\\\\n(a) For $i = 1, \\\\\\\\dots, n$, $a_i \\\\\\\\in S$.\\\\n(b) For $i,j = 1, \\\\\\\\dots, n$ (not necessarily distinct), $a_i - a_j \\\\\\\\in S$.\\\\n(c) For any integers $x,y \\\\\\\\in S$, if $x + y \\\\\\\\in S$, then $x - y \\\\\\\\in S$.\\\\nProve that $S$ must be equal to the set of all integers.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Let $ABC$ be a triangle with $\\\\\\\\angle ABC$ obtuse. The $A$-excircle is a circle in the exterior of $\\\\\\\\triangle ABC$ that is tangent to side $\\\\\\\\overline{BC}$ of the triangle and tangent to the extensions of the other two sides. Let $E$, $F$ be the feet of the altitudes from $B$ and $C$ to lines $AC$ and $AB$, respectively. Can line $EF$ be tangent to the $A$-excircle?\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $q = \\\\\\\\dfrac{3p-5}{2}$ where $p$ is an odd prime, and let\\\\n\\\\n\\\\\\\\[S_q =    \\\\\\\\frac{1}{2\\\\\\\\cdot 3 \\\\\\\\cdot 4} +   \\\\\\\\frac{1}{5\\\\\\\\cdot 6 \\\\\\\\cdot 7} + \\\\\\\\cdots +   \\\\\\\\frac{1}{q\\\\\\\\cdot (q+1) \\\\\\\\cdot (q+2)}.\\\\\\\\]\\\\n\\\\nProve that if $\\\\\\\\dfrac{1}{p}-2S_q = \\\\\\\\dfrac{m}{n}$ for integers\\\\n$m$ and $n$, then $m-n$ is divisible by $p$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"The repeating decimal $0.ab\\\\\\\\cdots k\\\\\\\\overline{pq\\\\\\\\cdots u}=\\\\\\\\frac mn$, where $m$ and $n$ are relatively prime integers, and there is at least one decimal before the repeating part. Show that $n$ is divisble by 2 or 5 (or both). (For example, $0.011\\\\\\\\overline{36}=0.01136363636\\\\\\\\cdots=\\\\\\\\frac 1{88}$, and 88 is divisible by 2.)\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"A sequence of functions $\\\\\\\\{f_n(x) \\\\\\\\}$ is defined recursively as follows:\\\\n\\\\\\\\begin{align*} f_1(x) &= \\\\\\\\sqrt {x^2 + 48}, \\\\\\\\quad \\\\\\\\text{and} \\\\\\\\\\\\\\\\\\\\nf_{n + 1}(x) &= \\\\\\\\sqrt {x^2 + 6f_n(x)} \\\\\\\\quad \\\\\\\\text{for } n \\\\\\\\geq 1. \\\\\\\\end{align*}\\\\n(Recall that $\\\\\\\\sqrt {}$ is understood to represent the positive square root.) For each positive integer $n$, find all real solutions of the equation $f_n(x) = 2x$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $q = \\dfrac{3p-5}{2}$ where $p$ is an odd prime, and let\n",
      "\n",
      "\n",
      "\\[S_q =    \\frac{1}{2\\cdot 3 \\cdot 4} +   \\frac{1}{5\\cdot 6 \\cdot 7} + \\cdots +   \\frac{1}{q\\cdot (q+1) \\cdot (q+2)}.\\]\n",
      "\n",
      "\n",
      "Prove that if $\\dfrac{1}{p}-2S_q = \\dfrac{m}{n}$ for integers\n",
      "$m$ and $n$, then $m-n$ is divisible by $p$.\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "The repeating decimal $0.ab\\cdots k\\overline{pq\\cdots u}=\\frac mn$, where $m$ and $n$ are relatively prime integers, and there is at least one decimal before the repeating part. Show that $n$ is divisble by 2 or 5 (or both). (For example, $0.011\\overline{36}=0.01136363636\\cdots=\\frac 1{88}$, and 88 is divisible by 2.)\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"A frog is placed at the origin on the number line, and moves according to the following rule: in a given move, the frog advances to either the closest point with a greater integer coordinate that is a multiple of 3, or to the closest point with a greater integer coordinate that is a multiple of 13. A move sequence is a sequence of coordinates that correspond to valid moves, beginning with 0 and ending with 39. For example, $0, 3, 6, 13, 15, 26, 39$ is a move sequence. How many move sequences are possible for the frog?\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"(Zuming Feng) Determine all composite positive integers \\\\\\\\(n\\\\\\\\) for which it is possible to arrange all divisors of \\\\\\\\(n\\\\\\\\) that are greater than 1 in a circle so that no two adjacent divisors are relatively prime.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $\\\\\\\\triangle{ABC}$ be a non-equilateral, acute triangle with $\\\\\\\\angle A=60^{\\\\\\\\circ}$, and let $O$ and $H$ denote the circumcenter and orthocenter of $\\\\\\\\triangle{ABC}$, respectively.\\\\n(a) Prove that line $OH$ intersects both segments $AB$ and $AC$.\\\\n(b) Line $OH$ intersects segments $AB$ and $AC$ at $P$ and $Q$, respectively. Denote by $s$ and $t$ the respective areas of triangle $APQ$ and quadrilateral $BPQC$. Determine the range of possible values for $s/t$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Determine whether or not there are any positive integral solutions of the simultaneous equations \\\\n\\\\\\\\begin{align*} x_1^2 +x_2^2 +\\\\\\\\cdots +x_{1985}^2 & = y^3,\\\\\\\\\\\\\\\\ x_1^3 +x_2^3 +\\\\\\\\cdots +x_{1985}^3 & = z^2 \\\\\\\\end{align*}\\\\nwith distinct integers $x_1,x_2,\\\\\\\\cdots,x_{1985}$.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Gemini not happy.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $\\\\\\\\{X_n\\\\\\\\}$ and $\\\\\\\\{Y_n\\\\\\\\}$ denote two sequences of integers defined as follows:\\\\n\\\\n $X_0=1$, $X_1=1$, $X_{n+1}=X_n+2X_{n-1}$ $(n=1,2,3,\\\\\\\\dots),$\\\\n$Y_0=1$, $Y_1=7$, $Y_{n+1}=2Y_n+3Y_{n-1}$ $(n=1,2,3,\\\\\\\\dots)$.\\\\nThus, the first few terms of the sequences are:\\\\n\\\\n $X:1, 1, 3, 5, 11, 21, \\\\\\\\dots$,\\\\n $Y:1, 7, 17, 55, 161, 487, \\\\\\\\dots$.\\\\nProve that, except for the \\\\\"1\\\\\", there is no term which occurs in both sequences.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square?\n",
      "\n",
      "[asy] unitsize(10mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4;  pair A=(0,1), B=(1,0), C=(1+sqrt(2),0), D=(2+sqrt(2),1), E=(2+sqrt(2),1+sqrt(2)), F=(1+sqrt(2),2+sqrt(2)), G=(1,2+sqrt(2)), H=(0,1+sqrt(2));  draw(A--B--C--D--E--F--G--H--cycle); draw(A--D); draw(B--G); draw(C--F); draw(E--H);  [/asy]\n",
      "\n",
      "$\\textbf{(A)}\\ \\frac{\\sqrt{2} - 1}{2} \\qquad\\textbf{(B)}\\ \\frac{1}{4} \\qquad\\textbf{(C)}\\ \\frac{2 - \\sqrt{2}}{2} \\qquad\\textbf{(D)}\\ \\frac{\\sqrt{2}}{4} \\qquad\\textbf{(E)}\\ 2 - \\sqrt{2}$\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Let $\\\\\\\\, P(z) \\\\\\\\,$ be a polynomial with complex coefficients which is of degree $\\\\\\\\, 1992 \\\\\\\\,$ and has distinct zeros. Prove that there exist complex numbers $\\\\\\\\, a_1, a_2, \\\\\\\\ldots, a_{1992} \\\\\\\\,$ such that $\\\\\\\\, P(z) \\\\\\\\,$ divides the polynomial $\\\\\\\\left( \\\\\\\\cdots \\\\\\\\left( (z-a_1)^2 - a_2 \\\\\\\\right)^2 \\\\\\\\cdots - a_{1991} \\\\\\\\right)^2 - a_{1992}$.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $ABCD$ be a quadrilateral inscribed in circle $\\\\\\\\omega$ with $AC \\\\\\\\perp BD$. Let $E$ and $F$ be the reflections of $D$ over lines $BA$ and $BC$, respectively, and let $P$ be the intersection of lines $BD$ and $EF$. Suppose that the circumcircle of $\\\\\\\\triangle EPD$ meets $\\\\\\\\omega$ at $D$ and $Q$, and the circumcircle of $\\\\\\\\triangle FPD$ meets $\\\\\\\\omega$ at $D$ and $R$. Show that $EQ = FR$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $\\{X_n\\}$ and $\\{Y_n\\}$ denote two sequences of integers defined as follows:\n",
      "\n",
      " $X_0=1$, $X_1=1$, $X_{n+1}=X_n+2X_{n-1}$ $(n=1,2,3,\\dots),$\n",
      "$Y_0=1$, $Y_1=7$, $Y_{n+1}=2Y_n+3Y_{n-1}$ $(n=1,2,3,\\dots)$.\n",
      "Thus, the first few terms of the sequences are:\n",
      "\n",
      " $X:1, 1, 3, 5, 11, 21, \\dots$,\n",
      " $Y:1, 7, 17, 55, 161, 487, \\dots$.\n",
      "Prove that, except for the \"1\", there is no term which occurs in both sequences.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"A father, mother and son hold a family tournament, playing a two person board game with no ties. The tournament rules are:\\\\n(i) The weakest player chooses the first two contestants.\\\\n(ii) The winner of any game plays the next game against the person left out.\\\\n(iii) The first person to win two games wins the tournament.\\\\nThe father is the weakest player, the son the strongest, and it is assumed that any player\\'s probability of winning an individual game from another player does not change during the tournament. Prove that the father\\'s optimal strategy for winning the tournament is to play the first game with his wife.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Prove that if $n$ is not a multiple of $3$, then the angle $\\\\\\\\frac{\\\\\\\\pi}{n}$ can be trisected with ruler and compasses.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"(Gregory Galperin) A square grid on the Euclidean plane consists of all points $(m,n)$, where $m$ and $n$ are integers.  Is it possible to cover all grid points by an infinite family of discs with non-overlapping interiors if each disc in the family has radius at least 5?\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $\\, P(z) \\,$ be a polynomial with complex coefficients which is of degree $\\, 1992 \\,$ and has distinct zeros. Prove that there exist complex numbers $\\, a_1, a_2, \\ldots, a_{1992} \\,$ such that $\\, P(z) \\,$ divides the polynomial $\\left( \\cdots \\left( (z-a_1)^2 - a_2 \\right)^2 \\cdots - a_{1991} \\right)^2 - a_{1992}$.\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Prove that if $n$ is not a multiple of $3$, then the angle $\\frac{\\pi}{n}$ can be trisected with ruler and compasses.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Let $a, b, c \\\\\\\\geq 0$ and satisfy\\\\n\\\\n$a^2 + b^2 + c^2 + abc = 4.$ \\\\nShow that\\\\n\\\\n$0 \\\\\\\\le ab + bc + ca - abc \\\\\\\\leq 2.$\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"A computer screen shows a $98 \\\\\\\\times 98$ chessboard, colored in the usual way. One can select with a mouse any rectangle with sides on the lines of the chessboard and click the mouse button: as a result, the colors in the selected rectangle switch (black becomes white, white becomes black). Find, with proof, the minimum number of mouse clicks needed to make the chessboard all one color.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $a_1, a_2, a_3, \\\\\\\\cdots$ be a non-decreasing sequence of positive integers. For $m \\\\\\\\ge 1$, define $b_m = \\\\\\\\min\\\\\\\\{n: a_n \\\\\\\\ge m\\\\\\\\}$, that is, $b_m$ is the minimum value of $n$ such that $a_n \\\\\\\\ge m$. If $a_{19} = 85$, determine the maximum value of $a_1 + a_2 + \\\\\\\\cdots + a_{19} + b_1 + b_2 + \\\\\\\\cdots + b_{85}$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $A, B, C, D$ denote four points in space and $AB$ the distance between $A$ and $B$, and so on. Show that\\\\n\\\\\\\\[AC^2 + BD^2 + AD^2 + BC^2 \\\\\\\\ge AB^2 + CD^2.\\\\\\\\]\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Jo and Blair take turns counting from $1$ to one more than the last number said by the other person. Jo starts by saying , so Blair follows by saying . Jo then says , and so on. What is the $53^{\\text{rd}}$ number said?\n",
      "$\\textbf{(A)}\\ 2 \\qquad \\textbf{(B)}\\ 3 \\qquad \\textbf{(C)}\\ 5 \\qquad \\textbf{(D)}\\ 6 \\qquad \\textbf{(E)}\\ 8$\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"In triangle $ABC$, points $P,Q,R$ lie on sides $BC,CA,AB$ respectively. Let $\\\\\\\\omega_A$, $\\\\\\\\omega_B$, $\\\\\\\\omega_C$ denote the circumcircles of triangles $AQR$, $BRP$, $CPQ$, respectively. Given the fact that segment $AP$ intersects $\\\\\\\\omega_A$, $\\\\\\\\omega_B$, $\\\\\\\\omega_C$ again at $X,Y,Z$ respectively, prove that $YX/XZ=BP/PC$.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $A,B,C,D$ denote four points in space and $AB$ the distance between $A$ and $B$, and so on. Show that \n",
      "\\[AC^2+BD^2+AD^2+BC^2\\ge AB^2+CD^2.\\]\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "In triangle $ABC$, points $P,Q,R$ lie on sides $BC,CA,AB$ respectively.  Let $\\omega_A$, $\\omega_B$, $\\omega_C$ denote the circumcircles of triangles $AQR$, $BRP$, $CPQ$, respectively.  Given the fact that segment $AP$ intersects $\\omega_A$, $\\omega_B$, $\\omega_C$ again at $X,Y,Z$ respectively, prove that $YX/XZ=BP/PC$.\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "[asy] draw((0,0)--(1,0)--(1,4)--(0,4)--(0,0)--(0,1)--(-1,1)--(-1,2)); draw((-1,2)--(0,2)--(0,4)--(-1,4)--(-1,5)--(1,5)--(1,6)--(0,6)); draw((0,6)--(0,5)--(3,5)--(3,6)--(4,6)--(4,2)--(5,2)); draw((5,2)--(5,1)--(1,1)--(3,1)--(3,0)--(4,0)--(4,1)); draw((1,4)--(3,4)--(3,2)--(1,2)--(4,2)--(3,2)--(3,6)); draw((3,6)--(4,6)--(4,5)--(5,5)--(5,4)--(4,4)); [/asy]\n",
      "Four rectangular paper strips of length $10$ and width $1$ are put flat on a table and overlap perpendicularly as shown. How much area of the table is covered?\n",
      "$\\text{(A)}\\ 36 \\qquad  \\text{(B)}\\ 40 \\qquad  \\text{(C)}\\ 44 \\qquad  \\text{(D)}\\ 98 \\qquad  \\text{(E)}\\ 100$\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Prove that the zeros of\\\\n\\\\\\\\[x^5+ax^4+bx^3+cx^2+dx+e=0\\\\\\\\]\\\\ncannot all be real if $2a^2<5b$.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Inside square $ABCD$ (See figure) with sides of length $12$ inches, segment $AE$ is drawn where $E$ is the point on $DC$ which is $5$ inches from $D$. \n",
      "The perpendicular bisector of $AE$ is drawn and intersects $AE, AD$, and $BC$ at points $M, P$, and $Q$ respectively. The ratio of segment $PM$ to $MQ$ is\n",
      "$\\textbf{(A) }5:12\\qquad \\textbf{(B) }5:13\\qquad \\textbf{(C) }5:19\\qquad \\textbf{(D) }1:4\\qquad  \\textbf{(E) }5:21$\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Call a real-valued function $f$ very convex if\\\\n\\\\\\\\[\\\\\\\\frac {f(x) + f(y)}{2} \\\\\\\\ge f\\\\\\\\left(\\\\\\\\frac {x + y}{2}\\\\\\\\right) + |x - y|\\\\\\\\]\\\\nholds for all real numbers $x$ and $y$. Prove that no very convex function exists.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Prove that the zeros of\n",
      "\\[x^5+ax^4+bx^3+cx^2+dx+e=0\\]\n",
      "cannot all be real if $2a^2<5b$.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Some checkers placed on an $n \\\\\\\\times n$ checkerboard satisfy the following conditions:\\\\n(a) every square that does not contain a checker shares a side with one that does;\\\\n(b) given any pair of squares that contain checkers, there is a sequence of squares containing checkers, starting and ending with the given squares, such that every two consecutive squares of the sequence share a side.\\\\nProve that at least $(n^{2}-2)/3$ checkers have been placed on the board.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"If the sum of the lengths of the six edges of a trirectangular tetrahedron $PABC$ (i.e., $\\\\\\\\angle APB=\\\\\\\\angle BPC=\\\\\\\\angle CPA=90^o$) is $S$, determine its maximum volume.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"By a partition $\\\\\\\\pi$ of an integer $n \\\\\\\\ge 1,$ we mean here a representation of $n$ as a sum of one or more positive integers where the summands must be put in nondecreasing order. (E.g., if $n=4,$ then the partitions $\\\\\\\\pi$ are $1+1+1+1,$ $1+1+2,$ $1+3, 2+2,$ and $4$).\\\\nFor any partition $\\\\\\\\pi,$ define $A(\\\\\\\\pi)$ to be the number of $1$\\'s which appear in $\\\\\\\\pi,$ and define $B(\\\\\\\\pi)$ to be the number of distinct integers which appear in $\\\\\\\\pi$ (E.g., if $n=13$ and $\\\\\\\\pi$ is the partition $1+1+2+2+2+5,$ then $A(\\\\\\\\pi)=2$ and $B(\\\\\\\\pi) = 3$).\\\\nProve that, for any fixed $n,$ the sum of $A(\\\\\\\\pi)$ over all partitions of $\\\\\\\\pi$ of $n$ is equal to the sum of $B(\\\\\\\\pi)$ over all partitions of $\\\\\\\\pi$ of $n.$\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "By a partition $\\pi$ of an integer $n\\ge 1,$ we mean here a representation of $n$ as a sum of one or more positive integers where the summands must be put in nondecreasing order. (E.g., if $n=4,$ then the partitions $\\pi$ are $1+1+1+1,$ $1+1+2,$ $1+3, 2+2,$ and $4$).\n",
      "For any partition $\\pi,$ define $A(\\pi)$ to be the number of $1$'s which appear in $\\pi,$ and define $B(\\pi)$ to be the number of distinct integers which appear in $\\pi$ (E.g., if $n=13$ and $\\pi$ is the partition $1+1+2+2+2+5,$ then $A(\\pi)=2$ and $B(\\pi) = 3$).\n",
      "Prove that, for any fixed $n,$ the sum of $A(\\pi)$ over all partitions of $\\pi$ of $n$ is equal to the sum of $B(\\pi)$ over all partitions of $\\pi$ of $n.$\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $a_{1}, a_{2}, \\\\\\\\dots, a_{n}$ ($n > 3$) be real numbers such that\\\\n\\\\\\\\[a_{1} + a_{2} + \\\\\\\\cdots + a_{n} \\\\\\\\geq n \\\\\\\\qquad \\\\\\\\mbox{and} \\\\\\\\qquad a_{1}^{2} + a_{2}^{2} + \\\\\\\\cdots + a_{n}^{2} \\\\\\\\geq n^{2}.\\\\\\\\]Prove that $\\\\\\\\max(a_{1}, a_{2}, \\\\\\\\dots, a_{n}) \\\\\\\\geq 2$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $a_{1}, a_{2}, \\dots, a_{n}$ ($n > 3$) be real numbers such that\n",
      "\\[a_{1} + a_{2} + \\cdots + a_{n} \\geq n \\qquad \\mbox{and} \\qquad a_{1}^{2} + a_{2}^{2} + \\cdots + a_{n}^{2} \\geq n^{2}.\\]\n",
      "Prove that $\\max(a_{1}, a_{2}, \\dots, a_{n}) \\geq 2$.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"In the diagram, if points $A, B$ and $C$ are points of tangency, then $x$ equals:\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Determine (with proof) whether there is a subset $X$ of the integers with the following property: for any integer $n$ there is exactly one solution of $a + 2b = n$ with $a,b \\\\\\\\in X$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Opposite sides of a regular hexagon are $12$ inches apart. The length of each side, in inches, is\n",
      "$\\textbf{(A) }7.5\\qquad \\textbf{(B) }6\\sqrt{2}\\qquad \\textbf{(C) }5\\sqrt{2}\\qquad \\textbf{(D) }\\frac{9}{2}\\sqrt{3}\\qquad  \\textbf{(D) }\\frac{9}{2}\\sqrt{3}\\qquad \\textbf{(E) }4\\sqrt{3}$\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Let $ABC$ be an acute-angled triangle whose side lengths satisfy the inequalities $AB < AC < BC$. If point $I$ is the center of the inscribed circle of triangle $ABC$ and point $O$ is the center of the circumscribed circle, prove that line $IO$ intersects segments $AB$ and $BC$.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Which of the following represents the result when the figure shown below is rotated clockwise $120^\\\\\\\\circ$ about its center?\\\\n[asy] unitsize(6); draw(circle((0,0),5)); draw((-1,2.5)--(1,2.5)--(0,2.5+sqrt(3))--cycle); draw(circle((-2.5,-1.5),1)); draw((1.5,-1)--(3,0)--(4,-1.5)--(2.5,-2.5)--cycle); [/asy]\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $ABCD$ be a cyclic quadrilateral. Prove that $|AB - CD| + |AD - BC| \\\\\\\\geq 2|AC - BD|$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $a_0, \\\\\\\\dots, a_n$ be real numbers in the interval $\\\\\\\\left(0,\\\\\\\\frac {\\\\\\\\pi}{2}\\\\\\\\right)$ such that\\\\n\\\\\\\\[\\\\\\\\tan{\\\\\\\\left(a_0 - \\\\\\\\frac {\\\\\\\\pi}{4}\\\\\\\\right)} + \\\\\\\\tan{\\\\\\\\left(a_1 - \\\\\\\\frac {\\\\\\\\pi}{4}\\\\\\\\right)} + \\\\\\\\dots + \\\\\\\\tan{\\\\\\\\left(a_n - \\\\\\\\frac {\\\\\\\\pi}{4}\\\\\\\\right)}\\\\\\\\ge n - 1\\\\\\\\]\\\\nProve that $\\\\\\\\tan{\\\\\\\\left(a_0\\\\\\\\right)}\\\\\\\\tan{\\\\\\\\left(a_1\\\\\\\\right)}\\\\\\\\cdots \\\\\\\\tan{\\\\\\\\left(a_n\\\\\\\\right)}\\\\\\\\ge n^{n + 1}$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"If $3(4x+5\\\\\\\\pi)=P$ then $6(8x+10\\\\\\\\pi)=$\",\\n\"A\": \"2P\",\\n\"B\": \"4P\",\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $ABC$ be an acute-angled triangle whose side lengths satisfy the inequalities $AB < AC < BC$. If point $I$ is the center of the inscribed circle of triangle $ABC$ and point $O$ is the center of the circumscribed circle, prove that line $IO$ intersects segments $AB$ and $BC$.\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $ABCD$ be a cyclic quadrilateral. Prove that \\[|AB - CD| + |AD - BC| \\geq 2|AC - BD|.\\]\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $a_0,\\cdots a_n$ be real numbers in the interval $\\left(0,\\frac {\\pi}{2}\\right)$ such that\n",
      "\\[\\tan{\\left(a_0 - \\frac {\\pi}{4}\\right)} + \\tan{\\left(a_1 - \\frac {\\pi}{4}\\right)} + \\cdots + \\tan{\\left(a_n - \\frac {\\pi}{4}\\right)}\\ge n - 1\\]\n",
      "Prove that $\\tan{\\left(a_0\\right)}\\tan{\\left(a_1\\right)}\\cdots \\tan{\\left(a_n\\right)}\\ge n^{n + 1}$.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"The square in the first diagram \\\\\"rolls\\\\\" clockwise around the fixed regular hexagon until it reaches the bottom.  In which position will the solid triangle be in diagram $4$?\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"(Zuming Feng, Zhonghao Ye) Let $ABCD$ be a quadrilateral, and let $E$ and $F$ be points on sides $AD$ and $BC$, respectively, such that $\\\\\\\\frac{AE}{ED} = \\\\\\\\frac{BF}{FC}$. Ray $FE$ meets rays $BA$ and $CD$ at $S$ and $T$ respectively. Prove that the circumcircles of triangles $SAE$, $SBF$, $TCF$, and $TDE$ pass through a common point.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "(Zuming Feng, Zhonghao Ye) Let $ABCD$ be a quadrilateral, and let $E$ and $F$ be points on sides $AD$ and $BC$, respectively, such that $AE/ED = BF/FC$.  Ray $FE$ meets rays $BA$ and $CD$ at $S$ and $T$ respectively. Prove that the circumcircles of triangles $SAE$, $SBF$, $TCF$, and $TDE$ pass through a common point.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Each point in the plane is assigned a real number such that, for any triangle, the number at the center of its inscribed circle is equal to the arithmetic mean of the three numbers at its vertices. Prove that all points in the plane are assigned the same number.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "The word \"HELP\" in block letters is painted in black with strokes $1$ unit wide on a $5$ by $15$ rectangular white sign with dimensions as shown.  The area of the white portion of the sign, in square units, is\n",
      "[asy] unitsize(12); fill((0,0)--(0,5)--(1,5)--(1,3)--(2,3)--(2,5)--(3,5)--(3,0)--(2,0)--(2,2)--(1,2)--(1,0)--cycle,black); fill((4,0)--(4,5)--(7,5)--(7,4)--(5,4)--(5,3)--(7,3)--(7,2)--(5,2)--(5,1)--(7,1)--(7,0)--cycle,black); fill((8,0)--(8,5)--(9,5)--(9,1)--(11,1)--(11,0)--cycle,black); fill((12,0)--(12,5)--(15,5)--(15,2)--(13,2)--(13,0)--cycle,black); fill((13,3)--(14,3)--(14,4)--(13,4)--cycle,white); draw((0,0)--(15,0)--(15,5)--(0,5)--cycle);  [/asy]\n",
      "$\\text{(A)}\\ 30 \\qquad \\text{(B)}\\ 32 \\qquad \\text{(C)}\\ 34 \\qquad \\text{(D)}\\ 36 \\qquad \\text{(E)}\\ 38$\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"(Ricky Liu) Find all positive integers $n$ such that there are $k\\\\\\\\ge 2$ positive rational numbers $a_1, a_2, \\\\\\\\ldots, a_k$ satisfying $a_1 + a_2 + \\\\\\\\cdots + a_k = a_1\\\\\\\\cdot a_2\\\\\\\\cdots a_k = n$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Let $ABC$ be a triangle.  Find all points $P$ on segment $BC$ satisfying the following property:  If $X$ and $Y$ are the intersections of line $PA$ with the common external tangent lines of the circumcircles of triangles $PAB$ and $PAC$, then $\\\\\\\\left(\\\\\\\\frac{PA}{XY}\\\\\\\\right)^2+\\\\\\\\frac{PB\\\\\\\\cdot PC}{AB\\\\\\\\cdot AC}=1.$\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Each point in the plane is assigned a real number such that, for any triangle, the number at the center of its inscribed circle is equal to the arithmetic mean of the three numbers at its vertices. Prove that all points in the plane are assigned the same number.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Legs $L_1, L_2, L_3, L_4$ of a square table each have length $n$, where $n$ is a positive integer. For how many ordered 4-tuples $(k_1, k_2, k_3, k_4)$ of nonnegative integers can we cut a piece of length $k_i$ from the end of leg $L_i$ $(i = 1,2,3,4)$ and still have a stable table? (The table is stable if it can be placed so that all four of the leg ends touch the floor. Note that a cut leg of length 0 is permitted.)\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"There are $n$ students standing in a circle, one behind the other. The students have heights $h_1 < h_2 < \\\\\\\\ldots < h_n$. If a student with height $h_k$ is standing directly behind a student with height $h_{k-2}$ or less, the two students are permitted to switch places. Prove that it is not possible to make more than $\\\\\\\\binom{n}{3}$ such switches before reaching a position in which no further switches are possible.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Gemini not happy.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Determine each real root of\\\\n$x^4-(2 \\\\\\\\cdot 10^{10}+1)x^2-x+10^{20}+10^{10}-1=0$\\\\ncorrect to four decimal places.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"For each integer $n \\\\\\\\ge 2$, determine, with proof, which of the two positive real numbers $a$ and $b$ satisfying\\\\n\\\\\\\\[a^n=a+1,\\\\\\\\]\\\\\\\\[ b^{2n}=b+3a\\\\\\\\]is larger.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $p > 2$ be a prime and let $a,b,c,d$ be integers not divisible by $p$, such that\\\\n\\\\\\\\[\\\\\\\\left\\\\\\\\{ \\\\\\\\dfrac{ra}{p} \\\\\\\\right\\\\\\\\} + \\\\\\\\left\\\\\\\\{ \\\\\\\\dfrac{rb}{p} \\\\\\\\right\\\\\\\\} + \\\\\\\\left\\\\\\\\{ \\\\\\\\dfrac{rc}{p} \\\\\\\\right\\\\\\\\} + \\\\\\\\left\\\\\\\\{ \\\\\\\\dfrac{rd}{p} \\\\\\\\right\\\\\\\\} = 2\\\\\\\\]\\\\nfor any integer $r$ not divisible by $p$. Prove that at least two of the numbers $a+b$, $a+c$, $a+d$, $b+c$, $b+d$, $c+d$ are divisible by $p$.\\\\n(Note: $\\\\\\\\{x\\\\\\\\} = x - \\\\\\\\lfloor x \\\\\\\\rfloor$ denotes the fractional part of $x$.)\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between $0$ and $1$ will $20!^{}$ be the resulting product?\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"How many positive integers have exactly three proper divisors (positive integral divisors excluding itself), each of which is less than 50?\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $p > 2$ be a prime and let $a,b,c,d$ be integers not divisible by $p$, such that\n",
      "\\[\\left\\{ \\dfrac{ra}{p} \\right\\} + \\left\\{ \\dfrac{rb}{p} \\right\\} + \\left\\{ \\dfrac{rc}{p} \\right\\} + \\left\\{ \\dfrac{rd}{p} \\right\\} = 2\\]\n",
      "for any integer $r$ not divisible by $p$. Prove that at least two of the numbers $a+b$, $a+c$, $a+d$, $b+c$, $b+d$, $c+d$ are divisible by $p$.\n",
      "(Note: $\\{x\\} = x - \\lfloor x \\rfloor$ denotes the fractional part of $x$.)\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"A ship travels from point $A$ to point $B$ along a semicircular path, centered at Island $X$. Then it travels along a straight path from $B$ to $C$. Which of these graphs best shows the ship\\'s distance from Island $X$ as it moves along its course?\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $a$, $b$, $c$ be positive real numbers such that $a^2 + b^2 + c^2 + (a + b + c)^2 \\\\\\\\le 4$.  Prove that\\\\n\\\\\\\\[\\\\\\\\frac{ab + 1}{(a + b)^2} + \\\\\\\\frac{bc + 1}{(b + c)^2} + \\\\\\\\frac{ca + 1}{(c + a)^2} \\\\\\\\ge 3.\\\\\\\\]\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $a$, $b$, $c$ be positive real numbers such that $a^2 + b^2 + c^2 + (a + b + c)^2 \\le 4$.  Prove that\n",
      "\\[\\frac{ab + 1}{(a + b)^2} + \\frac{bc + 1}{(b + c)^2} + \\frac{ca + 1}{(c + a)^2} \\ge 3.\\]\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Let $ABCDEF$ be a convex hexagon satisfying $\\\\\\\\overline{AB} \\\\\\\\parallel \\\\\\\\overline{DE}$, $\\\\\\\\overline{BC} \\\\\\\\parallel \\\\\\\\overline{EF}$, $\\\\\\\\overline{CD} \\\\\\\\parallel \\\\\\\\overline{FA}$, and\\\\n\\\\\\\\[AB \\\\\\\\cdot DE = BC \\\\\\\\cdot EF = CD \\\\\\\\cdot FA.\\\\\\\\]Let $X$, $Y$, and $Z$ be the midpoints of $\\\\\\\\overline{AD}$, $\\\\\\\\overline{BE}$, and $\\\\\\\\overline{CF}$. Prove that the circumcenter of $\\\\\\\\triangle ACE$, the circumcenter of $\\\\\\\\triangle BDF$, and the orthocenter of $\\\\\\\\triangle XYZ$ are collinear.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $ABCDEF$ be a convex hexagon satisfying $\\overline{AB} \\parallel \\overline{DE}$, $\\overline{BC} \\parallel \\overline{EF}$, $\\overline{CD} \\parallel \\overline{FA}$, and\\[AB \\cdot DE = BC \\cdot EF = CD \\cdot FA.\\]Let $X$, $Y$, and $Z$ be the midpoints of $\\overline{AD}$, $\\overline{BE}$, and $\\overline{CF}$. Prove that the circumcenter of $\\triangle ACE$, the circumcenter of $\\triangle BDF$, and the orthocenter of $\\triangle XYZ$ are collinear.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"The letter F shown below is rotated $90^\\\\\\\\circ$ clockwise around the origin, then reflected in the $y$-axis, and then rotated a half turn around the origin. What is the final image?\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"The insphere of a tetrahedron touches each face at its centroid. Show that the tetrahedron is regular.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Which one of the following bar graphs could represent the data from the circle graph?\\\\n[asy] unitsize(36); draw(circle((0,0),1),gray); fill((0,0)--arc((0,0),(0,-1),(1,0))--cycle,gray); fill((0,0)--arc((0,0),(1,0),(0,1))--cycle,black); [/asy]\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $S$ be a set with six elements. Let $\\\\\\\\mathcal{P}$ be the set of all subsets of $S.$ Subsets $A$ and $B$ of $S$, not necessarily distinct, are chosen independently and at random from $\\\\\\\\mathcal{P}$. The probability that $B$ is contained in one of $A$ or $S-A$ is $\\\\\\\\frac{m}{n^{r}},$ where $m$, $n$, and $r$ are positive integers, $n$ is prime, and $m$ and $n$ are relatively prime. Find $m+n+r.$ (The set $S-A$ is the set of all elements of $S$ which are not in $A.$)\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "The insphere of a tetrahedron touches each face at its centroid. Show that the tetrahedron is regular.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $a,b,c,d$ be real numbers such that $b-d \\\\\\\\ge 5$ and all zeros $x_1, x_2, x_3,$ and $x_4$ of the polynomial $P(x)=x^4+ax^3+bx^2+cx+d$ are real. Find the smallest value the product $(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)$ can take.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"An ordered pair $(m,n)$ of non-negative integers is called \\\\\"simple\\\\\" if the addition $m+n$ in base $10$ requires no carrying. Find the number of simple ordered pairs of non-negative integers that sum to $1492$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Let $ABC$ be a triangle such that\\\\n\\\\n$\\\\\\\\left( \\\\\\\\cot \\\\\\\\frac{A}{2} \\\\\\\\right)^2 + \\\\\\\\left( 2 \\\\\\\\cot \\\\\\\\frac{B}{2} \\\\\\\\right)^2 + \\\\\\\\left( 3 \\\\\\\\cot \\\\\\\\frac{C}{2} \\\\\\\\right)^2 = \\\\\\\\left( \\\\\\\\frac{6s}{7r} \\\\\\\\right)^2$,\\\\n\\\\nwhere $s$ and $r$ denote its semiperimeter and inradius, respectively.  Prove that triangle $ABC$ is similar to a triangle $T$ whose side lengths are all positive integers with no common divisor and determine those integers.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Failed extracting answer\n",
      "Prove that there is a constant $c>0$ with the following property: If $a, b, n$ are positive integers such that $\\gcd(a+i, b+j)>1$ for all $i, j\\in\\{0, 1, \\ldots n\\}$, then\\[\\min\\{a, b\\}>c^n\\cdot n^{\\frac{n}{2}}.\\]\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Determine all the roots, real or complex, of the system of simultaneous equations\\\\n\\\\n$x+y+z=3$,\\\\n$x^2+y^2+z^2=3$,\\\\n\\\\n$x^3+y^3+z^3=3$.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"A convex polyhedron has for its faces 12 squares, 8 regular hexagons, and 6 regular octagons. At each vertex of the polyhedron one square, one hexagon, and one octagon meet. How many segments joining vertices of the polyhedron lie in the interior of the polyhedron rather than along an edge or a face?\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Square $ABCD$ in the coordinate plane has vertices at the points $A(1,1), B(-1,1), C(-1,-1),$ and $D(1,-1).$ Consider the following four transformations:\n",
      "$\\quad\\bullet\\qquad$ $L,$ a rotation of $90^{\\circ}$ counterclockwise around the origin;\n",
      "$\\quad\\bullet\\qquad$ $R,$ a rotation of $90^{\\circ}$ clockwise around the origin;\n",
      "$\\quad\\bullet\\qquad$ $H,$ a reflection across the $x$-axis; and\n",
      "$\\quad\\bullet\\qquad$ $V,$ a reflection across the $y$-axis.\n",
      "Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying $R$ and then $V$ would send the vertex $A$ at $(1,1)$ to $(-1,-1)$ and would send the vertex $B$ at $(-1,1)$ to itself. How many sequences of $20$ transformations chosen from $\\{L, R, H, V\\}$ will send all of the labeled vertices back to their original positions? (For example, $R, R, V, H$ is one sequence of $4$ transformations that will send the vertices back to their original positions.)\n",
      "$\\textbf{(A)}\\ 2^{37} \\qquad\\textbf{(B)}\\ 3\\cdot 2^{36} \\qquad\\textbf{(C)}\\ 2^{38} \\qquad\\textbf{(D)}\\ 3\\cdot 2^{37} \\qquad\\textbf{(E)}\\ 2^{39}$\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "A [convex](https://artofproblemsolving.com/wiki/index.php/Convex) [polyhedron](https://artofproblemsolving.com/wiki/index.php/Polyhedron) has for its [faces](https://artofproblemsolving.com/wiki/index.php/Face) 12 [squares](https://artofproblemsolving.com/wiki/index.php/Square), 8 [regular](https://artofproblemsolving.com/wiki/index.php/Regular_polygon) [hexagons](https://artofproblemsolving.com/wiki/index.php/Hexagon), and 6 regular [octagons](https://artofproblemsolving.com/wiki/index.php/Octagon). At each [vertex](https://artofproblemsolving.com/wiki/index.php/Vertex) of the polyhedron one square, one hexagon, and one octagon meet. How many [segments](https://artofproblemsolving.com/wiki/index.php/Segment) joining vertices of the polyhedron lie in the interior of the polyhedron rather than along an [edge](https://artofproblemsolving.com/wiki/index.php/Edge) or a [face](https://artofproblemsolving.com/wiki/index.php/Face)?\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Let $a$, $b$, $c$ be positive real numbers.  Prove that\\\\n\\\\\\\\[\\\\\\\\frac{a^3 + 3b^3}{5a + b} + \\\\\\\\frac{b^3 + 3c^3}{5b + c} + \\\\\\\\frac{c^3 + 3a^3}{5c + a} \\\\\\\\ge \\\\\\\\frac{2}{3} (a^2 + b^2 + c^2).\\\\\\\\]\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Integers $n$ and $k$ are given, with $n \\\\\\\\ge k \\\\\\\\ge 2.$ You play the following game against an evil wizard.\\\\nThe wizard has $2n$ cards; for each $i = 1, ..., n,$ there are two cards labeled $i.$ Initially, the wizard places all cards face down in a row, in unknown order.\\\\nYou may repeatedly make moves of the following form: you point to any $k$ of the cards. The wizard then turns those cards face up. If any two of the cards match, the game is over and you win. Otherwise, you must look away, while the wizard arbitrarily permutes the $k$ chosen cards and turns them back face-down. Then, it is your turn again.\\\\nWe say this game is $\\\\\\\\textit{winnable}$ if there exist some positive integer $m$ and some strategy that is guaranteed to win in at most $m$ moves, no matter how the wizard responds.\\\\nFor which values of $n$ and $k$ is the game winnable?\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Each set of a finite family of subsets of a line is a union of two closed intervals. Moreover, any three of the sets of the family have a point in common. Prove that there is a point which is common to at least half the sets of the family.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSINGRetrying answer fetching:\n",
      "'```json\\n{\\n\"problem\": \"For a point $P = (a, a^2)$ in the coordinate plane, let $\\\\\\\\ell(P)$ denote the line passing through $P$ with slope $2a$. Consider the set of triangles with vertices of the form $P_1 = (a_1,  a_1^2)$, $P_2 = (a_2, a_2^2)$, $P_3 = (a_3, a_3^2)$, such that the intersections of the lines $\\\\\\\\ell(P_1)$, $\\\\\\\\ell(P_2)$, $\\\\\\\\ell(P_3)$ form an equilateral triangle $\\\\\\\\triangle$. Find the locus of the center of $\\\\\\\\triangle$ as $P_1P_2P_3$ ranges over all such triangles.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Each set of a finite family of subsets of a line is a union of two closed intervals. Moreover, any three of the sets of the family have a point in common. Prove that there is a point which is common to at least half the sets of the family.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"At the vertices of a regular hexagon are written six nonnegative integers whose sum is 2003. Bert is allowed to make moves of the following form: he may pick a vertex and replace the number written there by the absolute value of the difference between the numbers written at the two neighboring vertices. Prove that Bert can make a sequence of moves, after which the number 0 appears at all six vertices.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $K$ be the set of all positive integers that do not contain the digit $7$ in their base-$10$ representation. Find all polynomials $f$ with nonnegative integer coefficients such that $f(n) \\\\\\\\in K$ whenever $n \\\\\\\\in K$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "At the vertices of a regular hexagon are written six nonnegative integers whose sum is 2003. Bert is allowed to make moves of the following form: he may pick a vertex and replace the number written there by the absolute value of the difference between the numbers written at the two neighboring vertices. Prove that Bert can make a sequence of moves, after which the number 0 appears at all six vertices.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $\\\\\\\\mathcal{C}_1$ and $\\\\\\\\mathcal{C}_2$ be concentric circles, with $\\\\\\\\mathcal{C}_2$ in the interior of $\\\\\\\\mathcal{C}_1$. From a point $A$ on $\\\\\\\\mathcal{C}_1$ one draws the tangent $AB$ to $\\\\\\\\mathcal{C}_2$ ($B \\\\\\\\in \\\\\\\\mathcal{C}_2$). Let $C$ be the second point of intersection of $AB$ and $\\\\\\\\mathcal{C}_1$, and let $D$ be the midpoint of $AB$. A line passing through $A$ intersects $\\\\\\\\mathcal{C}_2$ at $E$ and $F$ in such a way that the perpendicular bisectors of $DE$ and $CF$ intersect at a point $M$ on $AB$. Find, with proof, the ratio $AM/MC$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Let $p_1, p_2, p_3, \\\\\\\\dots$ be the prime numbers listed in increasing order, and let $x_0$ be a real number between $0$ and $1$. For positive integer $k$, define\\\\n$x_{k}=\\\\\\\\begin{cases}0&\\\\\\\\text{ if }x_{k-1}=0\\\\\\\\\\\\\\\\ \\\\\\\\left\\\\\\\\{\\\\\\\\frac{p_{k}}{x_{k-1}}\\\\\\\\right\\\\\\\\}&\\\\\\\\text{ if }x_{k-1}\\\\\\\\ne0\\\\\\\\end{cases}$\\\\nwhere $\\\\\\\\{x\\\\\\\\}$ denotes the fractional part of $x$. (The fractional part of $x$ is given by $x-\\\\\\\\lfloor{x}\\\\\\\\rfloor$ where $\\\\\\\\lfloor{x}\\\\\\\\rfloor$ is the greatest integer less than or equal to $x$.) Find, with proof, all $x_0$ satisfying $0 < x_0 < 1$ for which the sequence $x_0, x_1, x_2, \\\\\\\\dots$ eventually becomes $0$.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Determine all integral solutions of $a^2+b^2+c^2=a^2b^2$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"$\\\\\\\\\\\\\\\\$Suppose that each square of a $4\\\\\\\\times 7$ chessboard, as shown above, is colored either black or white. Prove that with any such coloring, the board must contain a rectangle (formed by the horizontal and vertical lines of the board such as the one outlined in the figure) whose four distinct unit corner squares are all of the same color.\\\\\\\\\\\\\\\\\\\\nExhibit a black-white coloring of a $4\\\\\\\\times 6$ board in which the four corner squares of every rectangle, as described above, are not all of the same color.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "[asy] void fillsq(int x, int y){    fill((x,y)--(x+1,y)--(x+1,y+1)--(x,y+1)--cycle, mediumgray); } int i; fillsq(1,0);fillsq(4,0);fillsq(6,0); fillsq(0,1);fillsq(1,1);fillsq(2,1);fillsq(4,1);fillsq(5,1); fillsq(0,2);fillsq(2,2);fillsq(4,2); fillsq(0,3);fillsq(1,3);fillsq(4,3);fillsq(5,3); for(i=0; i<=7; ++i){draw((i,0)--(i,4),black+0.5);} for(i=0; i<=4; ++i){draw((0,i)--(7,i),black+0.5);} draw((3,1)--(3,3)--(7,3)--(7,1)--cycle,black+1); [/asy]\n",
      "\n",
      "(a) Suppose that each square of a $4\\times 7$ chessboard, as shown above, is colored either black or white. Prove that with any such coloring, the board must contain a rectangle (formed by the horizontal and vertical lines of the board such as the one outlined in the figure) whose four distinct unit corner squares are all of the same color.\n",
      "(b) Exhibit a black-white coloring of a $4\\times 6$ board in which the four corner squares of every rectangle, as described above, are not all of the same color.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Amanda Reckonwith draws five circles with radii $1, 2, 3, 4$ and $5$. Then for each circle she plots the point $(C,A)$, where $C$ is its circumference and $A$ is its area. Which of the following could be her graph?\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Consider an open interval of length $1/n$ on the real number line, where $n$ is a positive integer. Prove that the number of irreducible fractions $p/q$, with $1 \\\\\\\\le q \\\\\\\\le n$, contained in the given interval is at most $(n+1)/2$.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $P_1, \\\\\\\\ldots, P_{2n}$ be $2n$ distinct points on the unit circle $x^2 + y^2 = 1$ other than $(1,0)$. Each point is colored either red or blue, with exactly $n$ of them red and exactly $n$ of them blue. Let $R_1, \\\\\\\\ldots, R_n$ be any ordering of the red points. Let $B_1$ be the nearest blue point to $R_1$ traveling counterclockwise around the circle starting from $R_1$. Then let $B_2$ be the nearest of the remaining blue points to $R_2$ traveling counterclockwise around the circle from $R_2$, and so on, until we have labeled all the blue points $B_1, \\\\\\\\ldots, B_n$. Show that the number of counterclockwise arcs of the form $R_i \\\\\\\\rightarrow B_i$ that contain the point $(1,0)$ is independent of the way we chose the ordering $R_1, \\\\\\\\ldots, R_n$ of the red points.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"P, A, B, C, and D are five distinct points in space such that $\\\\\\\\angle APB = \\\\\\\\angle BPC = \\\\\\\\angle CPD = \\\\\\\\angle DPA = \\\\\\\\theta$, where $\\\\\\\\theta$ is a given acute angle. Determine the greatest and least values of $\\\\\\\\angle APC + \\\\\\\\angle BPD$.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Consider an open interval of length $1/n$ on the real number line, where $n$ is a positive integer. Prove that the number of irreducible fractions $p/q$, with $1\\le q\\le n$, contained in the given interval is at most $(n+1)/2$.\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $P_1, \\ldots, P_{2n}$ be $2n$ distinct points on the unit circle $x^2 + y^2 = 1$ other than $(1,0)$. Each point is colored either red or blue, with exactly $n$ of them red and exactly $n$ of them blue. Let $R_1, \\ldots, R_n$ be any ordering of the red points. Let $B_1$ be the nearest blue point to $R_1$ traveling counterclockwise around the circle starting from $R_1$. Then let $B_2$ be the nearest of the remaining blue points to $R_2$ traveling counterclockwise around the circle from $R_2$, and so on, until we have labeled all the blue points $B_1, \\ldots, B_n$. Show that the number of counterclockwise arcs of the form $R_i \\rightarrow B_i$ that contain the point $(1,0)$ is independent of the way we chose the ordering $R_1, \\ldots, R_n$ of the red points.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Let \\\\\\\\(ABC\\\\\\\\) be a triangle and let \\\\\\\\(\\\\\\\\omega\\\\\\\\) be its incircle. Denote by \\\\\\\\(D_1\\\\\\\\) and \\\\\\\\(E_1\\\\\\\\) the points where \\\\\\\\(\\\\\\\\omega\\\\\\\\) is tangent to sides \\\\\\\\(BC\\\\\\\\) and \\\\\\\\(AC\\\\\\\\), respectively. Denote by \\\\\\\\(D_2\\\\\\\\) and \\\\\\\\(E_2\\\\\\\\) the points on sides \\\\\\\\(BC\\\\\\\\) and \\\\\\\\(AC\\\\\\\\), respectively, such that \\\\\\\\(CD_2 = BD_1\\\\\\\\) and \\\\\\\\(CE_2 = AE_1\\\\\\\\), and denote by \\\\\\\\(P\\\\\\\\) the point of intersection of segments \\\\\\\\(AD_2\\\\\\\\) and \\\\\\\\(BE_2\\\\\\\\). Circle \\\\\\\\(\\\\\\\\omega\\\\\\\\) intersects segment \\\\\\\\(AD_2\\\\\\\\) at two points, the closer of which to the vertex \\\\\\\\(A\\\\\\\\) is denoted by \\\\\\\\(Q\\\\\\\\). Prove that \\\\\\\\(AQ = D_2P\\\\\\\\).\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Quadrilateral $XABY$ is inscribed in the semicircle $\\\\\\\\omega$ with diameter $XY$. Segments $AY$ and $BX$ meet at $P$. Point $Z$ is the foot of the perpendicular from $P$ to line $XY$. Point $C$ lies on $\\\\\\\\omega$ such that line $XC$ is perpendicular to line $AZ$. Let $Q$ be the intersection of segments $AY$ and $XC$. Prove that $\\\\\\\\dfrac{BY}{XP} + \\\\\\\\dfrac{CY}{XQ} = \\\\\\\\dfrac{AY}{AX}$.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $P$ be a point in the plane of triangle $ABC$ such that the segments $PA$, $PB$, and $PC$ are the sides of an obtuse triangle. Assume that in this triangle the obtuse angle opposes the side congruent to $PA$. Prove that $\\\\\\\\angle BAC$ is acute.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "A\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $ABC$ be an equilateral triangle and let $P$ be a point on its circumcircle. Let lines $PA$ and $BC$ intersect at $D$; let lines $PB$ and $CA$ intersect at $E$; and let lines $PC$ and $AB$ intersect at $F$. Prove that the area of triangle $DEF$ is twice that of triangle $ABC$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"$\\\\\\\\Delta ABC$ is a triangle with incenter $I$. Show that the circumcenters of $\\\\\\\\Delta IAB$, $\\\\\\\\Delta IBC$, and $\\\\\\\\Delta ICA$ lie on a circle whose center is the circumcenter of $\\\\\\\\Delta ABC$.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Given positive integers $m$ and $n$, prove that there is a positive integer $c$ such that the numbers $cm$ and $cn$ have the same number of occurrences of each non-zero digit when written in base ten.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $P$ be a point in the plane of triangle $ABC$ such that the segments $PA$, $PB$, and $PC$ are the sides of an obtuse triangle. Assume that in this triangle the obtuse angle opposes the side congruent to $PA$. Prove that $\\angle BAC$ is acute.\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "FAIL PARSINGFailed extracting answer\n",
      "\n",
      "($*$) Let $ABC$ be an equilateral triangle and let $P$ be a point on its circumcircle. Let lines $PA$ and $BC$ intersect at $D$; let lines $PB$ and $CA$ intersect at $E$; and let lines $PC$ and $AB$ intersect at $F$. Prove that the area of triangle $DEF$ is twice that of triangle $ABC$.\n",
      "'```json\\n{\\n\"problem\": \"Let $D$ be an arbitrary point on side $AB$ of a given triangle $ABC$, and let $E$ be the interior point where $CD$ intersects the external common tangent to the incircles of triangles $ACD$ and $BCD$. As $D$ assumes all positions between $A$ and $B$, prove that the point $E$ traces the arc of a circle.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "$\\Delta ABC$ is a triangle with incenter $I$. Show that the circumcenters of $\\Delta IAB$, $\\Delta IBC$, and $\\Delta ICA$ lie on a circle whose center is the circumcenter of $\\Delta ABC$.\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Given positive integers $m$ and $n$, prove that there is a positive integer $c$ such that the numbers $cm$ and $cn$ have the same number of occurrences of each non-zero digit when written in base ten.\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "The equation $z(z+i)(z+3i)=2002i$ has a zero of the form $a+bi$, where $a$ and $b$ are positive real numbers. Find $a.$\n",
      "$\\text{(A) }\\sqrt{118} \\qquad \\text{(B) }\\sqrt{210} \\qquad \\text{(C) }2 \\sqrt{210} \\qquad \\text{(D) }\\sqrt{2002} \\qquad \\text{(E) }100 \\sqrt{2}$\n",
      "FAIL PARSING\n",
      "Gemini not happy.'```json\\n{\\n\"problem\": \"For each positive integer $n$, let\\\\n\\\\\\\\begin{align*} S_n &= 1 + \\\\\\\\frac 12 + \\\\\\\\frac 13 + \\\\\\\\cdots + \\\\\\\\frac 1n \\\\\\\\\\\\\\\\\\\\nT_n &= S_1 + S_2 + S_3 + \\\\\\\\cdots + S_n \\\\\\\\\\\\\\\\\\\\nU_n &= \\\\\\\\frac{T_1}{2} + \\\\\\\\frac{T_2}{3} + \\\\\\\\frac{T_3}{4} + \\\\\\\\cdots + \\\\\\\\frac{T_n}{n+1}. \\\\\\\\end{align*}\\\\nFind, with proof, integers $0 < a, b, c, d < 1000000$ such that $T_{1988} = a S_{1989} - b$ and $U_{1988} = c S_{1989} - d$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"In triangle $ABC$, points $P,Q,R$ lie on sides $BC,CA,AB$ respectively.  Let $\\\\\\\\omega_A$, $\\\\\\\\omega_B$, $\\\\\\\\omega_C$ denote the circumcircles of triangles $AQR$, $BRP$, $CPQ$, respectively.  Given the fact that segment $AP$ intersects $\\\\\\\\omega_A$, $\\\\\\\\omega_B$, $\\\\\\\\omega_C$ again at $X,Y,Z$ respectively, prove that $YX/XZ=BP/PC$.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Prove that the average of the numbers $n\\\\\\\\sin(n^{\\\\\\\\circ})$ $(n = 2,4,6,\\\\\\\\ldots,180)$ is $\\\\\\\\cot(1^\\\\\\\\circ)$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Find all integers $n \\\\\\\\ge 3$ such that among any $n$ positive real numbers $a_1$, $a_2$, $\\\\\\\\dots$, $a_n$ with\\\\n\\\\\\\\[\\\\\\\\max(a_1, a_2, \\\\\\\\dots, a_n) \\\\\\\\le n \\\\\\\\cdot \\\\\\\\min(a_1, a_2, \\\\\\\\dots, a_n),\\\\\\\\]there exist three that are the side lengths of an acute triangle.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"(Reid Barton) An animal with $n$ cells is a connected figure consisting of $n$ equal-sized square cells.\\\\n\\\\nThe figure below shows an 8-cell animal.\\\\n\\\\n[asy]\\\\nunitsize(0.3inch);\\\\ndraw((0,0)--(1,0)--(1,1)--(0,1)--cycle);\\\\ndraw((1,0)--(2,0)--(2,1)--(1,1));\\\\ndraw((2,0)--(3,0)--(3,1)--(2,1));\\\\ndraw((1,1)--(1,2)--(0,2)--(0,1));\\\\ndraw((1,2)--(2,2)--(2,1));\\\\ndraw((2,1)--(2,2)--(3,2)--(3,1));\\\\ndraw((2,2)--(2,3)--(3,3)--(3,2));\\\\n[/asy]\\\\nA dinosaur is an animal with at least 2007 cells.  It is said to be primitive if its cells cannot be partitioned into two or more dinosaurs.  Find with proof the maximum number of cells in a primitive dinosaur.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "In triangle $ABC$, points $P,Q,R$ lie on sides $BC,CA,AB$ respectively.  Let $\\omega_A$, $\\omega_B$, $\\omega_C$ denote the circumcircles of triangles $AQR$, $BRP$, $CPQ$, respectively.  Given the fact that segment $AP$ intersects $\\omega_A$, $\\omega_B$, $\\omega_C$ again at $X,Y,Z$ respectively, prove that $YX/XZ=BP/PC$.\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Prove that the average of the numbers $n\\sin n^{\\circ}\\; (n = 2,4,6,\\ldots,180)$ is $\\cot 1^\\circ$.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Let $ABCD$ be an isosceles trapezoid with $AB \\\\\\\\parallel CD$. The inscribed circle $\\\\\\\\omega$ of triangle $BCD$ meets $CD$ at $E$. Let $F$ be a point on the (internal) angle bisector of $\\\\\\\\angle DAC$ such that $EF \\\\\\\\perp CD$. Let the circumscribed circle of triangle $ACF$ meet line $CD$ at $C$ and $G$. Prove that the triangle $AFG$ is isosceles.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree $n$ with real coefficients is the average of two monic polynomials of degree $n$ with $n$ real roots.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "A\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Prove that any monic polynomial (a polynomial with leading coefficient 1) of degree $n$ with real coefficients is the average of two monic polynomials of degree $n$ with $n$ real roots.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Prove that, for all positive real numbers $a, b, c,$ $\\\\\\\\frac{1}{a^3+b^3+abc}+\\\\\\\\frac{1}{b^3+c^3+abc}+\\\\\\\\frac{1}{a^3+c^3+abc} \\\\\\\\le \\\\\\\\frac{1}{abc}$.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Let $ABC$ be a scalene triangle with circumcircle $\\\\\\\\Omega$ and incenter $I.$ Ray $AI$ meets $BC$ at $D$ and $\\\\\\\\Omega$ again at $M;$ the circle with diameter $DM$ cuts $\\\\\\\\Omega$ again at $K.$ Lines $MK$ and $BC$ meet at $S,$ and $N$ is the midpoint of $IS.$ The circumcircles of $\\\\\\\\triangle KID$ and $\\\\\\\\triangle MAN$ intersect at points $L_1$ and $L.$ Prove that $\\\\\\\\Omega$ passes through the midpoint of either $IL_1$ or $IL.$\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"The five pieces shown below can be arranged to form four of the five figures shown in the choices. Which figure cannot be formed?\\\\n[asy] defaultpen(linewidth(0.6)); size(80); real r=0.5, s=1.5; path p=origin--(1,0)--(1,1)--(0,1)--cycle; draw(p); draw(shift(s,r)*p); draw(shift(s,-r)*p); draw(shift(2s,2r)*p); draw(shift(2s,0)*p); draw(shift(2s,-2r)*p); draw(shift(3s,3r)*p); draw(shift(3s,-3r)*p); draw(shift(3s,r)*p); draw(shift(3s,-r)*p); draw(shift(4s,-4r)*p); draw(shift(4s,-2r)*p); draw(shift(4s,0)*p); draw(shift(4s,2r)*p); draw(shift(4s,4r)*p); [/asy]\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Prove that, for all positive real numbers $a, b, c,$\n",
      "$\\frac{1}{a^3+b^3+abc}+\\frac{1}{b^3+c^3+abc}+\\frac{1}{a^3+c^3+abc} \\le \\frac{1}{abc}$.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Show that for any positive real $x$, $[nx] \\\\\\\\ge \\\\\\\\sum_{k=1}^{n} \\\\\\\\frac{[kx]}{k}$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"A calculator is broken so that the only keys that still work are the $\\\\\\\\sin, \\\\\\\\cos, \\\\\\\\tan, \\\\\\\\sin^{-1}, \\\\\\\\cos^{-1},$ and $\\\\\\\\tan^{-1}$ buttons. The display initially shows 0. Given any positive rational number $q$, show that pressing some finite sequence of buttons will yield $q$. Assume that the calculator does real number calculations with infinite precision. All functions are in terms of radians.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Show that for any positive real $x$, $[nx]\\ge \\sum_{1}^{n}\\left(\\frac{[kx]}{k}\\right)$\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"If ${a_1, a_2, a_3, ..., a_n}$ is a set of real numbers, indexed so that $a_1 < a_2 < a_3 < \\\\\\\\cdots < a_n,$ its complex power sum is defined to be $a_1 i + a_2 i^2 + a_3 i^3 + \\\\\\\\cdots + a_n i^n,$ where $i^2 = -1.$ Let $S_n$ be the sum of the complex power sums of all nonempty subsets of ${1, 2, ..., n}.$ Given that $S_8 = -176 - 64i$ and $S_9 = p + qi,$ where $p$ and $q$ are integers, find $|p| + |q|.$\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "At a gathering of $30$ people, there are $20$ people who all know each other and $10$ people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur?\n",
      "$\\textbf{(A)}\\ 240\\qquad\\textbf{(B)}\\ 245\\qquad\\textbf{(C)}\\ 290\\qquad\\textbf{(D)}\\ 480\\qquad\\textbf{(E)}\\ 490$\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Each cell of an $m \\\\\\\\times n$ board is filled with some nonnegative integer. Two numbers in the filling are said to be adjacent if their cells share a common side. (Note that two numbers in cells that share only a corner are not adjacent). The filling is called a garden if it satisfies the following two conditions:\\\\n(i) The difference between any two adjacent numbers is either $0$ or $1$. \\\\n(ii) If a number is less than or equal to all of its adjacent numbers, then it is equal to  $0$ .\\\\nDetermine the number of distinct gardens in terms of $m$ and $n$ .\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"During a certain lecture, each of five mathematicians fell asleep exactly twice. For each pair of mathematicians, there was some moment when both were asleep simultaneously. Prove that, at some moment, three of them were sleeping simultaneously.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "During a certain lecture, each of five mathematicians fell asleep exactly twice. For each pair of mathematicians, there was some moment when both were asleep simultaneously. Prove that, at some moment, three of them were sleeping simultaneously.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"(Titu Andreescu, Gabriel Dospinescu) For $m$ a positive integer, let $s(m)$ be the sum of the digits of $m$. For $n\\\\\\\\ge 2$, let $f(n)$ be the minimal $k$ for which there  exists a set $S$ of $n$ positive integers such that  $s\\\\\\\\left(\\\\\\\\sum_{x\\\\\\\\in X} x\\\\\\\\right) = k$ for any nonempty subset $X\\\\\\\\subset S$.  Prove that there are constants $0 < C_1 < C_2$ with\\\\n\\\\\\\\[C_1 \\\\\\\\log_{10} n \\\\\\\\le f(n) \\\\\\\\le C_2 \\\\\\\\log_{10} n.\\\\\\\\]\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"A word is defined as any finite string of letters.  A word is a palindrome if it reads the same backwards as forwards.  Let a sequence of words $W_0$, $W_1$, $W_2$, $\\\\\\\\dots$ be defined as follows: $W_0 = a$, $W_1 = b$, and for $n \\\\\\\\ge 2$, $W_n$ is the word formed by writing $W_{n - 2}$ followed by $W_{n - 1}$.  Prove that for any $n \\\\\\\\ge 1$, the word formed by writing $W_1$, $W_2$, $\\\\\\\\dots$, $W_n$ in succession is a palindrome.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"A positive integer is called ascending if, in its decimal representation, there are at least two digits and each digit is less than any digit to its right. How many ascending positive integers are there?\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "At the beginning of the school year, $50\\%$ of all students in Mr. Well's class answered \"Yes\" to the question \"Do you love math\", and $50\\%$ answered \"No.\" At the end of the school year, $70\\%$ answered \"Yes\" and $30\\%$ answered \"No.\" Altogether, $x\\%$ of the students gave a different answer at the beginning and end of the school year. What is the difference between the maximum and the minimum possible values of $x$?\n",
      "$\\textbf{(A)}\\ 0 \\qquad \\textbf{(B)}\\ 20 \\qquad \\textbf{(C)}\\ 40 \\qquad \\textbf{(D)}\\ 60 \\qquad \\textbf{(E)}\\ 80$\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "A word is defined as any finite string of letters.  A word is a palindrome if it reads the same backwards as forwards.  Let a sequence of words $W_0$, $W_1$, $W_2$, $\\dots$ be defined as follows: $W_0 = a$, $W_1 = b$, and for $n \\ge 2$, $W_n$ is the word formed by writing $W_{n - 2}$ followed by $W_{n - 1}$.  Prove that for any $n \\ge 1$, the word formed by writing $W_1$, $W_2$, $\\dots$, $W_n$ in succession is a palindrome.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Find, with proof, the least integer $N$ such that if any $2016$ elements are removed from the set $\\\\\\\\{1, 2,...,N\\\\\\\\}$, one can still find $2016$ distinct numbers among the remaining elements with sum $N$.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Prove that there are infinitely many distinct pairs $(a,b)$ of relatively prime positive integers $a>1$ and $b>1$ such that $a^b+b^a$ is divisible by $a+b$.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"(a) Do there exist 14 consecutive positive integers each of which is divisible by one or more primes $p$ from the interval $2 \\\\\\\\le p \\\\\\\\le 11$?\\\\n(b) Do there exist 21 consecutive positive integers each of which is divisible by one or more primes $p$ from the interval $2 \\\\\\\\le p \\\\\\\\le 13$?\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Point $P$ is $9$ units from the center of a circle of radius $15$. How many different chords of the circle contain $P$ and have integer lengths?\n",
      "(A) 11  (B) 12  (C) 13  (D) 14  (E) 29\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Prove that for any positive integer $k,$\\\\n\\\\\\\\[\\\\\\\\left(k^2\\\\\\\\right)!\\\\\\\\cdot\\\\\\\\prod_{j=0}^{k-1}\\\\\\\\frac{j!}{\\\\\\\\left(j+k\\\\\\\\right)!}\\\\\\\\]\\\\nis an integer.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Let $ABC$ be an acute triangle with $\\\\\\\\omega$, $\\\\\\\\Omega$, and $R$ being its incircle, circumcircle, and circumradius, respectively. Circle $\\\\\\\\omega_A$ is tangent internally to $\\\\\\\\Omega$ at $A$ and tangent externally to $\\\\\\\\omega$. Circle $\\\\\\\\Omega_A$ is tangent internally to $\\\\\\\\Omega$ at $A$ and tangent internally to $\\\\\\\\omega$. Let $P_A$ and $Q_A$ denote the centers of $\\\\\\\\omega_A$ and $\\\\\\\\Omega_A$, respectively. Define points $P_B$, $Q_B$, $P_C$, $Q_C$ analogously. Prove that\\\\n\\\\\\\\[8P_AQ_A \\\\\\\\cdot P_BQ_B \\\\\\\\cdot P_CQ_C \\\\\\\\le R^3,\\\\\\\\]with equality if and only if triangle $ABC$ is equilateral.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"What is the least common multiple of $\\\\\\\\frac{1}{x}$, $\\\\\\\\frac{1}{2x}$, and $\\\\\\\\frac{1}{3x}$?\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Prove that for any positive integer $k,$\n",
      "\\[\\left(k^2\\right)!\\cdot\\prod_{j=0}^{k-1}\\frac{j!}{\\left(j+k\\right)!}\\]\n",
      "is an integer.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Are there any triples $(a,b,c)$ of positive integers such that $(a-2)(b-2)(c-2) + 12$ is prime that properly divides the positive number $a^2 + b^2 + c^2 + abc - 2017$?\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Let $P$ be a given point inside quadrilateral $ABCD$. Points $Q_1$ and $Q_2$ are located within $ABCD$ such that $\\\\\\\\angle Q_1 BC = \\\\\\\\angle ABP$, $\\\\\\\\angle Q_1 CB = \\\\\\\\angle DCP$, $\\\\\\\\angle Q_2 AD = \\\\\\\\angle BAP$, $\\\\\\\\angle Q_2 DA = \\\\\\\\angle CDP$. Prove that $\\\\\\\\overline{Q_1 Q_2} \\\\\\\\parallel \\\\\\\\overline{AB}$ if and only if $\\\\\\\\overline{Q_1 Q_2} \\\\\\\\parallel \\\\\\\\overline{CD}$.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "What is the least common multiple of ${\\frac{1}{x}}$, $\\frac{1}{2x}$, and $\\frac{1}{3x}$ is $\\frac{1}{6x}$?\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "A large sphere is on a horizontal field on a sunny day. At a certain time the shadow of the sphere reaches out a distance \n",
      "of $10$ m from the point where the sphere touches the ground. At the same instant a meter stick \n",
      "(held vertically with one end on the ground) casts a shadow of length $2$ m. What is the radius of the sphere in meters? \n",
      "(Assume the sun's rays are parallel and the meter stick is a line segment.)\n",
      "$\\textbf{(A)}\\ \\frac{5}{2}\\qquad \\textbf{(B)}\\ 9 - 4\\sqrt{5}\\qquad \\textbf{(C)}\\ 8\\sqrt{10} - 23\\qquad \\textbf{(D)}\\ 6-\\sqrt{15}\\qquad \\textbf{(E)}\\ 10\\sqrt{5}-20$\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Suppose the sequence of nonnegative integers $a_1, a_2, ..., a_{1997}$ satisfies\\\\n$a_i + a_j \\\\\\\\le a_{i+j} \\\\\\\\le a_i + a_j + 1$\\\\nfor all $i, j \\\\\\\\ge 1$ with $i+j \\\\\\\\le 1997$. Show that there exists a real number $x$ such that $a_n = \\\\\\\\lfloor nx \\\\\\\\rfloor$ (the greatest integer $\\\\\\\\le nx$) for all $1 \\\\\\\\le n \\\\\\\\le 1997$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $P$ be a given point inside quadrilateral $ABCD$.  Points $Q_1$ and $Q_2$ are located within $ABCD$ such that $\\angle Q_1 BC = \\angle ABP$, $\\angle Q_1 CB = \\angle DCP$, $\\angle Q_2 AD = \\angle BAP$, $\\angle Q_2 DA = \\angle CDP$.  Prove that $\\overline{Q_1 Q_2} \\parallel \\overline{AB}$ if and only if $\\overline{Q_1 Q_2} \\parallel \\overline{CD}$.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"For any nonempty set $S$ of real numbers, let $\\\\\\\\sigma(S)$ denote the sum of the elements of $S$. Given a set $A$ of $n$ positive integers, consider the collection of all distinct sums $\\\\\\\\sigma(S)$ as $S$ ranges over the nonempty subsets of $A$. Prove that this collection of sums can be partitioned into $n$ classes so that in each class, the ratio of the largest sum to the smallest sum does not exceed 2.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Let $ABC$ be a triangle. Prove that there is a line $l$ (in the plane of triangle $ABC$) such that the intersection of the interior of triangle $ABC$ and the interior of its reflection $A\\'B\\'C\\'$ in $l$ has area more than $\\\\\\\\frac{2}{3}$ the area of triangle $ABC$.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Suppose the sequence of nonnegative integers $a_1,a_2,...,a_{1997}$ satisfies \n",
      "$a_i+a_j \\le a_{i+j} \\le a_i+a_j+1$\n",
      "for all $i, j \\ge 1$ with $i+j \\le 1997$. Show that there exists a real number $x$ such that $a_n=\\lfloor{nx}\\rfloor$ (the greatest integer $\\le nx$) for all $1 \\le n \\le 1997$.\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Seven cubes, whose volumes are $1$, $8$, $27$, $64$, $125$, $216$, and $343$ cubic units, are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except for the bottom cube, the bottom face of each cube lies completely on top of the cube below it. What is the total surface area of the tower (including the bottom) in square units?\n",
      "$\\textbf{(A)}\\ 644\\qquad\\textbf{(B)}\\ 658\\qquad\\textbf{(C)}\\ 664\\qquad\\textbf{(D)}\\ 720\\qquad\\textbf{(E)}\\ 749$\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"To clip a convex $n$-gon means to choose a pair of consecutive sides $AB, BC$ and to replace them by three segments $AM, MN,$ and $NC,$ where $M$ is the midpoint of $AB$ and $N$ is the midpoint of $BC$. In other words, one cuts off the triangle $MBN$ to obtain a convex $(n+1)$-gon. A regular hexagon $P_6$ of area $1$ is clipped to obtain a heptagon $P_7$. Then $P_7$ is clipped (in one of the seven possible ways) to obtain an octagon $P_8$, and so on. Prove that no matter how the clippings are done, the area of $P_n$ is greater than $\\\\\\\\frac{1}{3}$, for all $n\\\\\\\\ge6$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"A convex polygon $\\\\\\\\mathcal{P}$ in the plane is dissected into smaller convex polygons by drawing all of its diagonals. The lengths of all sides and all diagonals of the polygon $\\\\\\\\mathcal{P}$ are rational numbers. Prove that the lengths of all sides of all polygons in the dissection are also rational numbers.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Triangle $ABC$ is inscribed in a circle of radius $2$ with $\\\\\\\\angle ABC \\\\\\\\geq 90^\\\\\\\\circ$, and $x$ is a real number satisfying the equation $x^4 + ax^3 + bx^2 + cx + 1 = 0$, where $a=BC,b=CA,c=AB$. Find all possible values of $x$.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "To clip a convex $n$-gon means to choose a pair of consecutive sides $AB, BC$ and to replace them by three segments $AM, MN,$ and $NC,$ where $M$ is the midpoint of $AB$ and $N$ is the midpoint of $BC$. In other words, one cuts off the triangle $MBN$ to obtain a convex $(n+1)$-gon. A regular hexagon $P_6$ of area $1$ is clipped to obtain a heptagon $P_7$. Then $P_7$ is clipped (in one of the seven possible ways) to obtain an octagon $P_8$, and so on. Prove that no matter how the clippings are done, the area of $P_n$ is greater than $\\frac{1}{3}$, for all $n\\ge6$.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"(a) Prove that if the six dihedral (i.e. angles between pairs of faces) of a given tetrahedron are congruent, then the tetrahedron is regular.\\\\n(b) Is a tetrahedron necessarily regular if five dihedral angles are congruent?\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"A triangle is called a parabolic triangle if its vertices lie on a parabola $y = x^2$. Prove that for every nonnegative integer $n$, there is an odd number $m$ and a parabolic triangle with vertices at three distinct points with integer coordinates with area $(2^nm)^2$.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"Let $\\\\\\\\triangle ABC$ be an acute triangle, with $O$ as its circumcenter. Point $H$ is the foot of the perpendicular from $A$ to line $\\\\\\\\overleftrightarrow{BC}$, and points $P$ and $Q$ are the feet of the perpendiculars from $H$ to the lines $\\\\\\\\overleftrightarrow{AB}$ and $\\\\\\\\overleftrightarrow{AC}$, respectively.\\\\nGiven that $AH^2=2 \\\\\\\\cdot AO^2$, prove that the points $O,P,$ and $Q$ are collinear.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "A triangle is called a parabolic triangle if its vertices lie on a\n",
      "parabola $y = x^2$. Prove that for every nonnegative integer $n$, there\n",
      "is an odd number $m$ and a parabolic triangle with vertices at three\n",
      "distinct points with integer coordinates with area $(2^nm)^2$.\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $\\triangle ABC$ be an acute triangle, with $O$ as its circumcenter. Point $H$ is the foot of the perpendicular from $A$ to line $\\overleftrightarrow{BC}$, and points $P$ and $Q$ are the feet of the perpendiculars from $H$ to the lines $\\overleftrightarrow{AB}$ and $\\overleftrightarrow{AC}$, respectively.\n",
      "Given that \\[AH^2=2\\cdot AO^2,\\]prove that the points $O,P,$ and $Q$ are collinear.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Trapezoid $ABCD$, with $\\\\\\\\overline{AB} \\\\\\\\parallel \\\\\\\\overline{CD}$, is inscribed in circle $\\\\\\\\omega$ and point $G$ lies inside triangle $BCD$. Rays $AG$ and $BG$ meet $\\\\\\\\omega$ again at points $P$ and $Q$, respectively. Let the line through $G$ parallel to $\\\\\\\\overline{AB}$ intersect $\\\\\\\\overline{BD}$ and $\\\\\\\\overline{BC}$ at points $R$ and $S$, respectively. Prove that quadrilateral $PQRS$ is cyclic if and only if $\\\\\\\\overline{BG}$ bisects $\\\\\\\\angle CBD$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "D\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Trapezoid $ABCD$, with $\\overline{AB}||\\overline{CD}$, is inscribed in circle $\\omega$ and point $G$ lies inside triangle $BCD$.  Rays $AG$ and $BG$ meet $\\omega$ again at points $P$ and $Q$, respectively.  Let the line through $G$ parallel to $\\overline{AB}$ intersect $\\overline{BD}$ and $\\overline{BC}$ at points $R$ and $S$, respectively.  Prove that quadrilateral $PQRS$ is cyclic if and only if $\\overline{BG}$ bisects $\\angle CBD$.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"$\\\\\\\\text{A, B}$, and $\\\\\\\\text{C}$ are three interior points of a sphere $S$ such that $AB$ and $AC$ are perpendicular to the diameter of $S$ through $\\\\\\\\text{A}$, and so that two spheres can be constructed through $\\\\\\\\text{A}$, $\\\\\\\\text{B}$, and $\\\\\\\\text{C}$ which are both tangent to $S$. Prove that the sum of their radii is equal to the radius of $S$.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "The figure consists of alternating light and dark squares. The number of dark squares exceeds the number of light squares by\n",
      "$\\text{(A)}\\ 7 \\qquad \\text{(B)}\\ 8 \\qquad \\text{(C)}\\ 9 \\qquad \\text{(D)}\\ 10 \\qquad \\text{(E)}\\ 11$\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "$A,B$, and $C$ are three interior points of a sphere $S$ such that $AB$ and $AC$ are perpendicular to the diameter of $S$ through $A$, and so that two spheres can be constructed through $A$, $B$, and $C$ which are both tangent to $S$. Prove that the sum of their radii is equal to the radius of $S$.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Let $ABCD$ be a cyclic quadrilateral satisfying $AD^2 + BC^2 = AB^2$. The diagonals of $ABCD$ intersect at $E$. Let $P$ be a point on side $\\\\\\\\overline{AB}$ satisfying $\\\\\\\\angle APD = \\\\\\\\angle BPC$. Show that line $PE$ bisects $\\\\\\\\overline{CD}$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Nine mathematicians meet at an international conference and discover that among any three of them, at least two speak a common language. If each of the mathematicians speak at most three languages, prove that there are at least three of the mathematicians who can speak the same language.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "A\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Nine mathematicians meet at an international conference and discover that among any three of them, at least two speak a common language. If each of the mathematicians speak at most three languages, prove that there are at least three of the mathematicians who can speak the same language.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"A circle is divided into 432 congruent arcs by 432 points. The points are colored in four colors such that some 108 points are colored Red, some 108 points are colored Green, some 108 points are colored Blue, and the remaining 108 points are colored Yellow. Prove that one can choose three points of each color in such a way that the four triangles formed by the chosen points of the same color are congruent.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Mike leaves home and drives slowly east through city traffic. When he reaches the highway he drives east more rapidly until he reaches the shopping mall where he stops. He shops at the mall for an hour. Mike returns home by the same route as he came, driving west rapidly along the highway and then slowly through city traffic. Each graph shows the distance from home on the vertical axis versus the time elapsed since leaving home on the horizontal axis. Which graph is the best representation of Mike\\'s trip?\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance $d$ traveled by the two animals over time $t$ from start to finish?\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "A circle is divided into 432 congruent arcs by 432 points.  The points are colored in four colors such that some 108 points are colored Red, some 108 points are colored Green, some 108 points are colored Blue, and the remaining 108 points are colored Yellow.  Prove that one can choose three points of each color in such a way that the four triangles formed by the chosen points of the same color are congruent.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Six segments $S_1, S_2, S_3, S_4, S_5,$ and $S_6$ are given in a plane. These are congruent to the edges $AB, AC, AD, BC, BD,$ and $CD$, respectively, of a tetrahedron $ABCD$. Show how to construct a segment congruent to the altitude of the tetrahedron from vertex $A$ with straight-edge and compasses.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Six segments $S_1, S_2, S_3, S_4, S_5,$ and $S_6$ are given in a plane. These are congruent to the edges $AB, AC, AD, BC, BD,$ and $CD$, respectively, of a tetrahedron $ABCD$. Show how to construct a segment congruent to the altitude of the tetrahedron from vertex $A$ with straight-edge and compasses.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Jorge\\'s teacher asks him to plot all the ordered pairs $(w, l)$ of positive integers for which $w$ is the width and $l$ is the length of a rectangle with area 12. What should his graph look like?\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "Gemini not happy.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Prove that if $a$, $b$, and $c$ are positive real numbers, then\\\\n\\\\n$a^ab^bc^c\\\\\\\\ge (abc)^{(a+b+c)/3}$\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"(András Gyárfás) Let \\\\\\\\(S\\\\\\\\) be a set containing \\\\\\\\(n^2+n-1\\\\\\\\) elements, for some positive integer \\\\\\\\(n\\\\\\\\).  Suppose that the \\\\\\\\(n\\\\\\\\)-element subsets of \\\\\\\\(S\\\\\\\\) are partitioned into two classes.  Prove that there are at least \\\\\\\\(n\\\\\\\\) pairwise disjoint sets in the same class.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Let $S$ be a set of integers (not necessarily positive) such that\\\\n(a) there exist $a,b \\\\\\\\in S$ with $\\\\\\\\gcd(a,b) = \\\\\\\\gcd(a - 2,b - 2) = 1$;\\\\n(b) if $x$ and $y$ are elements of $S$ (possibly equal), then $x^2 - y$ also belongs to $S$. \\\\nProve that $S$ is the set of all integers.\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Prove that if $a$, $b$, and $c$ are positive real numbers, then\n",
      "\n",
      "$a^ab^bc^c\\ge (abc)^{(a+b+c)/3}$\n",
      "Retrying answer fetching:\n",
      "NO MULTIPLE CHOICE\n",
      "Failed extracting answer\n",
      "Let $S$ be a set of integers (not necessarily positive) such that\n",
      "(a) there exist $a,b \\in S$ with $\\gcd(a,b) = \\gcd(a - 2,b - 2) = 1$;\n",
      "(b) if $x$ and $y$ are elements of $S$ (possibly equal), then $x^2 - y$ also belongs to $S$. \n",
      "Prove that $S$ is the set of all integers.\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"Given circles $\\\\\\\\omega_1$ and $\\\\\\\\omega_2$ intersecting at points $X$ and $Y$, let $\\\\\\\\ell_1$ be a line through the center of $\\\\\\\\omega_1$ intersecting $\\\\\\\\omega_2$ at points $P$ and $Q$ and let $\\\\\\\\ell_2$ be a line through the center of $\\\\\\\\omega_2$ intersecting $\\\\\\\\omega_1$ at points $R$ and $S$.  Prove that if $P, Q, R$ and $S$ lie on a circle then the center of this circle lies on line $XY$.\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n\"problem\": \"The equation $2^{333x-2} + 2^{111x+2} = 2^{222x+1} + 1$ has three real roots. Given that their sum is $m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$\",\\n\"A\": null,\\n\"B\": null,\\n\"C\": null,\\n\"D\": null,\\n\"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"(Sam Vandervelde) Let $n$ be a positive integer. Define a sequence by setting $a_1 = n$ and, for each $k>1$, letting $a_k$ be the unique integer in the range $0 \\\\\\\\le a_k \\\\\\\\le k-1$ for which $a_1 + a_2 + \\\\\\\\cdots + a_k$ is divisible by $k$. For instance, when $n=9$ the obtained sequence is $9, 1, 2, 0, 3, 3, 3, \\\\\\\\ldots$. Prove that for any $n$ the sequence $a_1, a_2, a_3, \\\\\\\\ldots$ eventually becomes constant.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n \"problem\": \"Car M traveled at a constant speed for a given time. This is shown by the dashed line. Car N traveled at twice the speed for the same distance. If Car M and Car N\\'s speed and time are shown as solid line, which graph illustrates this?\",\\n \"A\": null,\\n \"B\": null,\\n \"C\": null,\\n \"D\": null,\\n \"E\": null\\n}\\n```'\n",
      "FAIL PARSING\n",
      "'```json\\n{\\n  \"problem\": \"A blackboard contains 68 pairs of nonzero integers. Suppose that for each positive integer $k$ at most one of the pairs $(k, k)$ and $(-k, -k)$ is written on the blackboard. A student erases some of the 136 integers, subject to the condition that no two erased integers may add to 0. The student then scores one point for each of the 68 pairs in which at least one integer is erased. Determine, with proof, the largest number $N$ of points that the student can guarantee to score regardless of which 68 pairs have been written on the board.\",\\n  \"A\": null,\\n  \"B\": null,\\n  \"C\": null,\\n  \"D\": null,\\n  \"E\": null\\n}\\n```'\n"
     ]
    }
   ],
   "source": [
    "import concurrent.futures\n",
    "\n",
    "# Use a ProcessPoolExecutor with up to 10 workers\n",
    "with concurrent.futures.ProcessPoolExecutor(max_workers=64) as executor:\n",
    "    # executor.map(...) applies process_entry to each item in `subset`.\n",
    "    # It returns results in the same order as `subset`.\n",
    "    results = list(executor.map(process_entry_no_mc, amc_dataset))\n",
    "\n",
    "    # Now `results` is a list of parsed dictionaries, one for each entry\n",
    "    final_list = []\n",
    "    counter =0\n",
    "    for entry, parsed_dict in zip(amc_dataset, results):\n",
    "        if parsed_dict:\n",
    "            if parsed_dict['problem'] is None:\n",
    "                continue\n",
    "            final_list.append(parsed_dict)\n",
    "            counter +=1\n",
    "            if counter%100==0:\n",
    "                # Save final list as json\n",
    "                with open(\"amc_processed.json\", \"w\") as f:\n",
    "                    json.dump(final_list, f, indent=2)\n",
    "# Save final list as json\n",
    "with open(\"amc_processed.json\", \"w\") as f:\n",
    "    json.dump(final_list, f, indent=2)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Finally, manually review dataset"
   ]
  }
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